Irisan Kerucut (Cone) – Rumus, Contoh Soal dan Jawaban

Pinter Pandai

6 tahun ago

Irisan Kerucut

Dalam matematika, irisan kerucut adalah lokus (sekumpulan titik-titik) dari semua titik yang membentuk kurva dua-dimensi, yang terbentuk oleh irisan sebuah kerucut dengan sebuah bidang.

3 Jenis kurva pada irisan kerucut

Tiga jenis kurva yang dapat terjadi adalah:

Apollonius dari Perga adalah matematikawan Yunani yang pertama mempelajari irisan kerucut secara sistematik pada awal abad ke-2 SM.

Suatu kerucut jika diiris horizontal, maka irisannya berbentuk lingkaran.

Jika kerucut tersebut dipotong secara miring (dan tidak memotong alasnya), maka terbentuk suatu elips.

Jika mengirisnya memotong alasnya dan memotongnya secara vertikal, maka terbentuk suatu hiperbola.

Jika mengirisnya memotong alasnya dan memotongnya tidak secara vertikal, maka terbentuk suatu parabola.

Geometri

Dalam memahami geometri irisan kerucut, sebuah kerucut dianggap memiliki dua kulit yang membentang sampai tak berhingga di kedua arah.

Sebuah generator adalah sebuah garis yang dapat dibuat pada kulit kerucut, dan semua generator saling berpotongan di satu titik yang disebut verteks kerucut.

Jenis-jenis irisan kerucut

Jika sebuah bidang mengiris kerucut sejajar dengan satu dan hanya satu generator, maka irisannya adalah parabola.
Jika bidang pengiris sejajar dengan dua generator, maka irisannya akan memotong kedua kulit dan membentuk sebuah hiperbola.
Sebuah elips terjadi jika bidang pengiris tidak sejajar dengan generator mana pun.
Lingkaran adalah kasus khusus dari elips, yang terbentuk jika bidang pengiris memotong semua generator dan tegak lurus sumbu kerucut.

Bagaimana kasus degenerasi dapat terjadi?

Kasus-kasus degenerasi terjadi jika bidang-bidang pengiris melalui verteks kerucut. Irisan-irisannya dapat berupa titik, garis lurus, dan dua garis lurus yang saling berpotongan.
Sebuah titik terjadi jika bidang pengiris melalui verteks kerucut namun tidak memotong generator mana pun. Kasus ini merupakan elips yang terdegenerasi.
Jika bidang pengiris melalui verteks kerucut, dan hanya satu generator, maka yang terjadi adalah sebuah garis lurus, dan merupakan parabola yang terdegenerasi.
Sebuah hiperbola terdegenerasi terjadi jika bidang pengiris melalui verteks kerucut dan dua generator sehingga memberikan dua garis lurus yang saling berpotongan.

Geometri analitis

Secara geometri analitis, irisan kerucut dapat didefinisikan sebagai:

“

tempat kedudukan titik-titik pada sebuah bidang, sedemikian, sehingga jarak titik-titik tersebut ke sebuah titik tetap F (yang disebut fokus) memiliki rasio yang konstan terhadap jarak titik-titik tersebut ke sebuah garis tetap L (disebut direktriks) yang tidak mengandung F.

”

Eksentrisitas adalah rasio antara FM dan M’M.Elips (e=1/2), parabola (e=1) dan hiperbola (e=2) dengan fokus (F) dan direktriks yang tetap.

Rasio yang konstan tersebut disebut eksentrisitas, dilambangkan dengan e, dan merupakan bilangan non-negatif. Untuk e = 0, irisan kerucut tersebut adalah lingkaran, 0 < e < 1 sebuah elips, e = 1 sebuah parabola, dan e > 1 sebuah hiperbola.

Koordinat Kartesius

Dalam koordinat kartesius, grafik dari persamaan kuadrat dengan dua variabel selalu menghasilkan irisan kerucut, dan semua irisan kerucut dapat dihasilkan dengan cara ini.

Jika terdapat persamaan dengan bentuk:

{<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/039cb6e3f95e5a24631538049f2e33d426e33f90" alt="{\displaystyle ax^{2}+2hxy+by^{2}+2gx+2fy+c=0}" aria-hidden="true" width="16570.7" height="1295.7">

maka:

Jika h² = ab, persamaan ini menghasilkan parabola.
Jika h² < ab, persamaan ini menghasilkan elips.
Jika h² > ab, persamaan ini menghasilkan hiperbola.
Jika a = b dan h = 0, persamaan ini menghasilkan lingkaran.
Jika a + b = 0, persamaan ini menghasilkan hiperbola persegi.

Bentuk persamaan umum irisan kerucut

Bentuk persamaan umum irisan kerucut sebagai berikut:

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a694f3a0805f2f11868a2a69413866d64170409b" alt="{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0}" aria-hidden="true" width="600" height="400">

kesimpulan:

Jika A = B = 0 maka persamaan adalah garis lurus/linear
Jika A = B = 0 tetapi tidak kedua-duanya maka persamaan adalah parabola/kuadrat
Jika A = B maka persamaan adalah lingkaran
Jika A ≠ B dan bertanda positif maka persamaan adalah elips
Jika A ≠ B dan bertanda negatif maka persamaan adalah hiperbola

Rumus irisan kerucut tentang: linkaran, parabola, elips dan hiperbola

Lingkaran: Titik pusat (0,0): $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37dd4f282df84a83620f71dc52345122e0e3a514" alt="{\displaystyle x^{2}+y^{2}=r^{2}}" aria-hidden="true" width="600" height="400">$; Titik pusat (h,k): $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73fbc024ba8bfa16cfd5d3415f89b66848e1cb4b" alt="{\displaystyle (x-h)^{2}+(y-k)^{2}=r^{2}}" aria-hidden="true" width="600" height="400">$ atau $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c26ef11fcac5371a75b212dfe481c0c7a459d9a" alt="{\displaystyle x^{2}+2hx+h^{2}+y^{2}+2ky+k^{2}-r^{2}=0}" aria-hidden="true" width="600" height="400">$

dengan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59038f0e411d37bf8e6f2d39e25a901919d666bd" alt="{\displaystyle x^{2}+y^{2}+Ax+By+C=0}" aria-hidden="true" width="600" height="400">$ maka $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a995a2d42470d0ea17720b8cfa97f0091482d36" alt="{\displaystyle A=2h,B=2ksertaC=h^{2}+k^{2}-r^{2}}" aria-hidden="true" width="600" height="400">$

Parabola

	Vertikal	Horisontal
	Titik pusat (0,0)
Persamaan	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3f7178d35b71078913a6dda45aa35c18b0d7d0" alt="{\displaystyle x^{2}=4py}" aria-hidden="true" width="600" height="400">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c380f1f2286c8b0f3ebaadad97959ad155488d9" alt="{\displaystyle y^{2}=4px}" aria-hidden="true" width="600" height="400">$
Sumbu simetri	sumbu y	sumbu x
Fokus	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51f1f48e74076ff928a4c92030ed41944201e5f2" alt="{\displaystyle F(0,p)}" aria-hidden="true" width="600" height="400">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/087d2607d7457c8cd8dd980f2921f24ad18735c3" alt="{\displaystyle F(p,0)}" aria-hidden="true" width="600" height="400">$
Direktris	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e9a2a0c0d50deaa16e19ff2ec86f3546c6f61d" alt="{\displaystyle y=-p}" aria-hidden="true" width="600" height="400">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a613b75dcd05b82819d921877993e318b12c17e2" alt="{\displaystyle x=-p}" aria-hidden="true" width="600" height="400">$
	Titik pusat (h,k)
Persamaan	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6557f42b456c85644e99bdcb2fec5feec410e1" alt="{\displaystyle (x-h)^{2}=4p(y-k)}" aria-hidden="true" width="600" height="400">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4132aabfcb6cafdd5171d4004798363f93c289da" alt="{\displaystyle (y-k)^{2}=4p(x-h)}" aria-hidden="true" width="600" height="400">$
Sumbu simetri	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d77df94024e4133b5b9cd4b387cfb6548d198d" alt="{\displaystyle x=h}" aria-hidden="true" width="600" height="400">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724459ad8c8f6908b575c4392af6f8b6cbc669e5" alt="{\displaystyle y=k}" aria-hidden="true" width="600" height="400">$
Fokus	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daa43bdd7ac9e3a94e49bbdbfb12bb49d28028cd" alt="{\displaystyle F(h,k+p)}" aria-hidden="true" width="600" height="400">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29564cac82830b45c8eb5c2d3dac4dedd2a3e046" alt="{\displaystyle F(h+p,k)}" aria-hidden="true" width="600" height="400">$
Direktris	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a86ace07bc705edd831f96c5ffef845f914f8cf" alt="{\displaystyle y=k-p}" aria-hidden="true" width="600" height="400">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/061f99b42b395f589c417239cbe06161529cd21a" alt="{\displaystyle x=h-p}" aria-hidden="true" width="600" height="400">$

Elips

dimana $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78a9d2e194c630274028a18c59e4289981aad14d" alt="{\displaystyle c={\sqrt {a^{2}-b^{2}}}}" aria-hidden="true" width="5857.8" height="1510.9">$

Hiperbola

dimana $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fedf2f36b06db4bed9920046bd74c332c99ea4a2" alt="{\displaystyle c={\sqrt {a^{2}+b^{2}}}}" aria-hidden="true" width="5857.8" height="1510.9">$

Persamaan garis singgung

bergradien $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" alt="{\displaystyle m}" aria-hidden="true" width="878.5" height="721.6">$ ( $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3daca9fbcf73b8a2b575a10e2d5de25f9f1c6f49" alt="{\displaystyle y=mx+c}" aria-hidden="true" width="4939" height="1008.6">$ )

	Vertikal	Horisontal
	Titik pusat (0,0)
Lingkaran	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faf10db6b779f2fc5588a5e0ce4345d27ec31eeb" alt="{\displaystyle y=mx\pm r{\sqrt {1+m^{2}}}}" aria-hidden="true" width="9013.4" height="1510.9">$
Parabala	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8ea5ba14098e94d0b2eb2ec79fb0400f368540" alt="{\displaystyle y=mx-pm}" aria-hidden="true" width="5887.5" height="1008.6">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9165d4b46166a40388fd2929f0574bef17090f" alt="{\displaystyle y=mx+{\frac {p}{m}}}" aria-hidden="true" width="5744" height="2085">$
Elips	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b60120060168dd5c346322fcfcee08f1a21932ce" alt="{\displaystyle y=mx\pm {\sqrt {b^{2}+a^{2}m^{2}}}}" aria-hidden="true" width="9928.2" height="1510.9">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd21b1d94f4e1c7fb916a71a38e63873c88e6c7" alt="{\displaystyle y=mx\pm {\sqrt {a^{2}m^{2}+b^{2}}}}" aria-hidden="true" width="9928.2" height="1510.9">$
Hiperbola	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/774122f12a569d646f7e774474616cdbbb88f99c" alt="{\displaystyle y=mx\pm {\sqrt {b^{2}-a^{2}m^{2}}}}" aria-hidden="true" width="9928.2" height="1510.9">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5518e6d286eb3498d52769ec3530148a283666f" alt="{\displaystyle y=mx\pm {\sqrt {a^{2}m^{2}-b^{2}}}}" aria-hidden="true" width="9928.2" height="1510.9">$
	Titik pusat (h,k)
Lingkaran	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae96bb1bfe282864d6466dea32a110b1ba98c40" alt="{\displaystyle (y-k)=m(x-h)\pm r{\sqrt {1+m}}}" aria-hidden="true" width="13494.3" height="1295.7">$
Parabala	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf94672d5ad556452385db5b7f0544fa23035ab" alt="{\displaystyle (y-k)=m(x-h)-pm}" aria-hidden="true" width="10989.4" height="1223.9">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c887b5a8697b7ef363c81a6c2c9e17efd220c9" alt="{\displaystyle (y-k)=m(x-h)+{\frac {p}{m}}}" aria-hidden="true" width="10845.9" height="2085">$
Elips	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d0d819ef0ff5979400fa4765cf59b5b85397bc" alt="{\displaystyle (y-k)=m(x-h)\pm {\sqrt {b^{2}+a^{2}m^{2}}}}" aria-hidden="true" width="15030.1" height="1510.9">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165b9aea874aa2df76aaf34c051d4c2f51e928ab" alt="{\displaystyle (y-k)=m(x-h)\pm {\sqrt {a^{2}m^{2}+b^{2}}}}" aria-hidden="true" width="15030.1" height="1510.9">$
Hiperbola	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/774122f12a569d646f7e774474616cdbbb88f99c" alt="{\displaystyle y=mx\pm {\sqrt {b^{2}-a^{2}m^{2}}}}" aria-hidden="true" width="9928.2" height="1510.9">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5518e6d286eb3498d52769ec3530148a283666f" alt="{\displaystyle y=mx\pm {\sqrt {a^{2}m^{2}-b^{2}}}}" aria-hidden="true" width="9928.2" height="1510.9">$

jika persamaan garis lurus bergradien sejajar maka

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a421d100f7813bdaa32bca9d049c3971985db028" alt="{\displaystyle m_{2}=m_{1}}" aria-hidden="true" width="3998.9" height="865.1">

jika persamaan garis lurus bergradien tegak lurus maka

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a543f99096dae7356c08090282fdfa97fc2397" alt="{\displaystyle m_{2}={\frac {-1}{m_{1}}}}" aria-hidden="true" width="4358.9" height="2372">

melalui titik $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" alt="{\displaystyle (x_{1},y_{1})}" aria-hidden="true" width="3195" height="1223.9">$

dengan cara bagi adil

	Vertikal	Horisontal
	Titik pusat (0,0)
Lingkaran	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/324f6a1b4981386dc392b67eed71524e50e23515" alt="{\displaystyle xx_{1}+yy_{1}=r^{2}}" aria-hidden="true" width="6503.2" height="1295.7">$
Parabola	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b398456ae0933d79799f8c7c2d420f2e1b35d57" alt="{\displaystyle xx_{1}=2py+2py_{1}}" aria-hidden="true" width="7605.8" height="1080.4">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1a095910b7fe5ddeb98acc98d1bc4aa09db6ec" alt="{\displaystyle yy_{1}=2px+2px_{1}}" aria-hidden="true" width="7605.8" height="1080.4">$
Elips	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/795b979095486b47497c699d17cf2b62bbe06106" alt="{\displaystyle {\frac {xx_{1}}{b^{2}}}+{\frac {yy_{1}}{a^{2}}}=1}" aria-hidden="true" width="6818.3" height="2228.5">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a752ec7728c2112d2ad7b4f7d13a5b556afcda0a" alt="{\displaystyle {\frac {xx_{1}}{a^{2}}}+{\frac {yy_{1}}{b^{2}}}=1}" aria-hidden="true" width="6818.3" height="2228.5">$
Hiperbola	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1c93ebe2ed2a6f45159f320f4a1cfab37b18a3" alt="{\displaystyle {\frac {xx_{1}}{b^{2}}}-{\frac {yy_{1}}{a^{2}}}=1}" aria-hidden="true" width="6818.3" height="2228.5">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/294b2e00414f975c6bd258c3e0e8f7b337137d09" alt="{\displaystyle {\frac {xx_{1}}{a^{2}}}-{\frac {yy_{1}}{b^{2}}}=1}" aria-hidden="true" width="6818.3" height="2228.5">$
	Titik pusat (h,k)
Lingkaran	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e98f83a93d05a4007962a79a06cd300db2cfdaaf" alt="{\displaystyle xx_{1}+yy_{1}+{\frac {1}{2}}Ax+{\frac {1}{2}}Ax_{1}+{\frac {1}{2}}By+{\frac {1}{2}}By_{1}+C=0}" aria-hidden="true" width="22476.3" height="2228.5">$
Parabola	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d692d8a9078168ff3ca07e0562650132fb4d7c" alt="{\displaystyle (x-h)(x_{1}-h)=2p(y-k)+2p(y_{1}-k)}" aria-hidden="true" width="17809.6" height="1223.9">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b52aab54e72fea3984f162a95e6d0834344bd407" alt="{\displaystyle (y-k)(y_{1}-k)=2p(x-h)+2p(x_{1}-h)}" aria-hidden="true" width="17809.6" height="1223.9">$
Elips	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df610d6ea7e76784dd1161232c3f9b75f8803115" alt="{\displaystyle {\frac {(x-h)(x_{1}-h)}{b^{2}}}+{\frac {(y-k)(y_{1}-k)}{a^{2}}}=1}" aria-hidden="true" width="17022.1" height="2587.3">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5d76061326ea234d11b4abc261a4bf29eaebe7" alt="{\displaystyle {\frac {(x-h)(x_{1}-h)}{a^{2}}}+{\frac {(y-k)(y_{1}-k)}{b^{2}}}=1}" aria-hidden="true" width="17022.1" height="2587.3">$
Hiperbola	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5308fc5273e8ea06e74111aa3b54b0e386c25304" alt="{\displaystyle {\frac {(x-h)(x_{1}-h)}{b^{2}}}-{\frac {(y-k)(y_{1}-k)}{a^{2}}}=1}" aria-hidden="true" width="17022.1" height="2587.3">$	$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f02bc4c90eaecfdcc0a3494b6b25252e369933f9" alt="{\displaystyle {\frac {(x-h)(x_{1}-h)}{a^{2}}}-{\frac {(y-k)(y_{1}-k)}{b^{2}}}=1}" aria-hidden="true" width="17022.1" height="2587.3">$

jika titik

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" alt="{\displaystyle (x_{1},y_{1})}" aria-hidden="true" width="3195" height="1223.9">

berada di dalam bentuknya maka ada 1 persamaan garis singgung (1 langkah).

jika titik

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" alt="{\displaystyle (x_{1},y_{1})}" aria-hidden="true" width="3195" height="1223.9">

berada di luar bentuknya maka ada 2 persamaan garis singgung (2 langkah).

Contoh Soal dan Jawaban Irisan Kerucut (Cone)

Titik pusat (0,0). Tentukan persamaan garis singgung yang bergradien 2 terhadap $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">$ !

jawab:

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8ef90b7c2f247a410da08703178812232328cb" alt="{\displaystyle y^{2}=16x-&gt;y^{2}=4(4x)jadip=4}" aria-hidden="true" width="14762.8" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c747d38ce039180082be9e0e41e6d1130c29abf5" alt="{\displaystyle y=2x+{\frac {4}{2}}-&gt;y=2x+2}" aria-hidden="true" width="11728.6" height="2228.5">

Tentukan persamaan garis singgung yang melalui (4,8) terhadap $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">$ !

jawab:

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c65ffdfd646c3b1e6422940808f3474b5b64c1f2" alt="{\displaystyle y^{2}-16x=0 (4,8)(8)^{2}-16(4)=0=0}" width="468" height="25" aria-hidden="true">

(dalam)

dengan cara bagi adil

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1a095910b7fe5ddeb98acc98d1bc4aa09db6ec" alt="{\displaystyle yy_{1}=2px+2px_{1}}" aria-hidden="true" width="7605.8" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c481954fe1dd85d7ef525309aa6b55921f82795f" alt="{\displaystyle 8y=8x+8(4)}" aria-hidden="true" width="6408" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/259a42d6217251f0074d4bfbb1b4cbfac093642e" alt="{\displaystyle 8y=8x+32}" aria-hidden="true" width="5629" height="1080.4">

(dibagi 8)

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c0942d12215c8d1816d3e5ab419a56863bee4bb" alt="{\displaystyle y=x+4}" aria-hidden="true" width="4127.5" height="1080.4">

Tentukan persamaan garis singgung yang melalui (1,5) terhadap $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">$ !

jawab:

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d62c72e0231ea74737edd1bea0fa6d0f0df1c8" alt="{\displaystyle y^{2}-16x=0makamasukkanlah(1,5)(5)^{2}-16(1)=9&gt;0}" aria-hidden="true" width="25202.2" height="1367.4">

(luar)

dengan cara bagi adil

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1a095910b7fe5ddeb98acc98d1bc4aa09db6ec" alt="{\displaystyle yy_{1}=2px+2px_{1}}" aria-hidden="true" width="7605.8" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c4735a02cc95d7d3ed614b078c621a1a3e87c91" alt="{\displaystyle 5y=8x+8(1)}" aria-hidden="true" width="6408" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eddf2d4da0697aa3c16268ebf70d87a2879dd55c" alt="{\displaystyle 5y=8x+8}" aria-hidden="true" width="5128.5" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47fd8ec896c3d2ccb3f0c68acb153a85bfdecbbd" alt="{\displaystyle y={\frac {8}{5}}x+{\frac {8}{5}}}" aria-hidden="true" width="5348" height="2228.5">

masukkan lah $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">$

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25d1e7f2dfb70f92a55db76bc0900816a4a90e6" alt="{\displaystyle ({\frac {8}{5}}x+{\frac {8}{5}})^{2}=16x}" aria-hidden="true" width="7656.9" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5a50b9dcbec984bacf56ca0db165e4cb60c873" alt="{\displaystyle {\frac {64}{25}}x^{2}+{\frac {128}{25}}x+{\frac {64}{25}}-16x=0}" aria-hidden="true" width="13259.3" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf334a0f37689758ff6aa6bf5dc7d1e8e45f0279" alt="{\displaystyle {\frac {64}{25}}x^{2}+{\frac {128}{25}}x+{\frac {64}{25}}-{\frac {400}{25}}x=0}" aria-hidden="true" width="14119.8" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e52cc91476972c5c4db517bd971b54c6e1789c29" alt="{\displaystyle {\frac {64}{25}}x^{2}-{\frac {272}{25}}x+{\frac {64}{25}}=0}" aria-hidden="true" width="10462.9" height="2228.5">

(dibagi 16/25)

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f885e8f44d5a2b1c5c34b2c4b70d1f85e1771b" alt="{\displaystyle 4x^{2}-17x+4=0}" aria-hidden="true" width="7881.4" height="1223.9">

maka kita mencari nilai x

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c22777378f9c594c71158fea8946f2495f2a28" alt="{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}" aria-hidden="true" width="9267.9" height="2659.1">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6495a84b9671f0fd8c6f5db2eb4823b0c0b2747c" alt="{\displaystyle x={\frac {17\pm {\sqrt {289-256}}}{8}}}" aria-hidden="true" width="9549.9" height="2515.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bfb8f59a8a68fa48c59e4f246d6aa268c9fe8f9" alt="{\displaystyle x={\frac {17\pm {\sqrt {33}}}{8}}}" aria-hidden="true" width="6325" height="2515.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4878439391ebacfbf757bc5a9ccf50afc6d92355" alt="{\displaystyle x_{1}={\frac {17+{\sqrt {33}}}{8}}}" aria-hidden="true" width="6778.9" height="2515.6">

atau

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d51f3cfc032e41d04919872756362ac1c6fb3e9" alt="{\displaystyle x_{2}={\frac {17-{\sqrt {33}}}{8}}}" aria-hidden="true" width="6778.9" height="2515.6">

maka kita mencari nilai y

untuk

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" alt="{\displaystyle x_{1}}" aria-hidden="true" width="1026.4" height="865.1">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e92b71ec31bca031b0fc81f767cbfdf550b38e" alt="{\displaystyle y_{1}={\frac {8}{5}}({\frac {17+{\sqrt {33}}}{8}})+{\frac {8}{5}}}" aria-hidden="true" width="10419.9" height="2515.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2c7c3e4485742207e3a683df23594ad31ca85c" alt="{\displaystyle y_{1}={\frac {17}{5}}+{\frac {\sqrt {33}}{5}}+{\frac {8}{5}}}" aria-hidden="true" width="9140.4" height="2515.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0499c9b0e0ef4e063fe06c9fb2d10aa008785c9c" alt="{\displaystyle y_{1}=5+{\frac {\sqrt {33}}{5}}}" aria-hidden="true" width="6196.4" height="2515.6">

jadi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d898e930701a765360ec56e79be79a085fb5b195" alt="{\displaystyle ({\frac {17+{\sqrt {33}}}{8}},5+{\frac {\sqrt {33}}{5}})}" aria-hidden="true" width="9560.6" height="2515.6">$

untuk

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" alt="{\displaystyle x_{2}}" aria-hidden="true" width="1026.4" height="865.1">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da41bb9c9bbbc22bcafe29f64ffd9beb692dd0b" alt="{\displaystyle y_{2}={\frac {8}{5}}({\frac {17-{\sqrt {33}}}{8}})+{\frac {8}{5}}}" aria-hidden="true" width="10419.9" height="2515.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d27d64e6e4f69b1c40816b39db250bd017f8ffb7" alt="{\displaystyle y_{2}={\frac {17}{5}}-{\frac {\sqrt {33}}{5}}+{\frac {8}{5}}}" aria-hidden="true" width="9140.4" height="2515.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30bee0156847ec7958c704a18bd2cd260d64eacf" alt="{\displaystyle y_{2}=5-{\frac {\sqrt {33}}{5}}}" aria-hidden="true" width="6196.4" height="2515.6">

jadi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67afe8f4ec427ac4d107d43f027b188cb6b8330a" alt="{\displaystyle ({\frac {17-{\sqrt {33}}}{8}},5-{\frac {\sqrt {33}}{5}})}" aria-hidden="true" width="9560.6" height="2515.6">$

kembali dengan cara bagi adil

untuk persamaan singgung pertama

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1a095910b7fe5ddeb98acc98d1bc4aa09db6ec" alt="{\displaystyle yy_{1}=2px+2px_{1}}" aria-hidden="true" width="7605.8" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c867187a1a12a8a58c17b171656146d63d16e1b9" alt="{\displaystyle (5+{\frac {\sqrt {33}}{5}})y=8x+8({\frac {17+{\sqrt {33}}}{8}})}" aria-hidden="true" width="14522.4" height="2515.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f751b37565b38cc507992b5ad9d5e948861d5cfe" alt="{\displaystyle (5+{\frac {\sqrt {33}}{5}})y=8x+17+{\sqrt {33}}}" aria-hidden="true" width="12882.9" height="2515.6">

untuk persamaan singgung kedua

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69020d92649c568586fc8b2b94f3896478ccd4ef" alt="{\displaystyle yy_{2}=2px+2px_{2}}" aria-hidden="true" width="7605.8" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48db56285dad0012b869e2e2ba3e576928940d11" alt="{\displaystyle (5-{\frac {\sqrt {33}}{5}})y=8x+8({\frac {17-{\sqrt {33}}}{8}})}" aria-hidden="true" width="14522.4" height="2515.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04a05aff41bcf9d41a5a13ead64be886d71b3b8" alt="{\displaystyle (5-{\frac {\sqrt {33}}{5}})y=8x+17-{\sqrt {33}}}" aria-hidden="true" width="12882.9" height="2515.6">

Titik pusat (h,k). Tentukan persamaan garis singgung $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2aa603dea12dddc98204dcecf8ec862b5d2d2b" alt="{\displaystyle y^{2}-6y-8x+9=0}" aria-hidden="true" width="9028.4" height="1295.7">$ melalui persamaan yang tegak lurus $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bd241a112277c6567ae8c2b41122b11b9322da" alt="{\displaystyle y-2x-5=0}" aria-hidden="true" width="6351.4" height="1080.4">$ !

jawab: ubah ke bentuk sederhana

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2aa603dea12dddc98204dcecf8ec862b5d2d2b" alt="{\displaystyle y^{2}-6y-8x+9=0}" aria-hidden="true" width="9028.4" height="1295.7">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee0e75cb0abf8ce78674ce26d29dd94465b45e3" alt="{\displaystyle y^{2}-6y+9=8x}" aria-hidden="true" width="7305" height="1295.7">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240df7e655e1e9a1d926e2b24d89a00e49c3cc17" alt="{\displaystyle (y-3)^{2}=8x}" aria-hidden="true" width="5860.9" height="1367.4">

cari gradien persamaan ${\displaystyle y-2x-5=0}<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bd241a112277c6567ae8c2b41122b11b9322da" alt="{\displaystyle y-2x-5=0}" aria-hidden="true" width="6351.4" height="1080.4">$

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bd241a112277c6567ae8c2b41122b11b9322da" alt="{\displaystyle y-2x-5=0}" aria-hidden="true" width="6351.4" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dcfd6c110b33eb69951410bda4e5576e56b2000" alt="{\displaystyle y=2x+5}" aria-hidden="true" width="4628" height="1080.4">

gradien ( $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31aafa60e48d39ccce922404c0b80340b2cc777a" alt="{\displaystyle m_{1}}" aria-hidden="true" width="1332.4" height="865.1">$ ) = 2 karena tegak lurus menjadi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aff0003dc00d7fb574c0b8fd1b2acb5b87157f7" alt="{\displaystyle m_{2}=-{\frac {1}{2}}}" aria-hidden="true" width="4305.5" height="2228.5">$

cari $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" alt="{\displaystyle p}" aria-hidden="true" width="542" height="865.1">$

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e62eec77dad92cb604a3ece6d31ead2b008698" alt="{\displaystyle (y-3)^{2}=8x-&gt;(y-3)^{2}=4(2x)jadip=2}" aria-hidden="true" width="19262.9" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6eac42016ff58db1ffae7c6e4ecc92d39c136d3" alt="{\displaystyle y=-{\frac {1}{2}}x+{\frac {2}{-{\frac {1}{2}}}}-&gt;y=-{\frac {1}{2}}x-4}" aria-hidden="true" width="14997.5" height="2874.4">

Tentukan persamaan garis singgung $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2aa603dea12dddc98204dcecf8ec862b5d2d2b" alt="{\displaystyle y^{2}-6y-8x+9=0}" aria-hidden="true" width="9028.4" height="1295.7">$ yang berordinat 6!

jawab: ubah ke bentuk sederhana

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2aa603dea12dddc98204dcecf8ec862b5d2d2b" alt="{\displaystyle y^{2}-6y-8x+9=0}" aria-hidden="true" width="9028.4" height="1295.7">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee0e75cb0abf8ce78674ce26d29dd94465b45e3" alt="{\displaystyle y^{2}-6y+9=8x}" aria-hidden="true" width="7305" height="1295.7">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240df7e655e1e9a1d926e2b24d89a00e49c3cc17" alt="{\displaystyle (y-3)^{2}=8x}" aria-hidden="true" width="5860.9" height="1367.4">

cari absis dimana ordinat 6

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240df7e655e1e9a1d926e2b24d89a00e49c3cc17" alt="{\displaystyle (y-3)^{2}=8x}" aria-hidden="true" width="5860.9" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a358bbe7b2e8919ad28d681ec5e1a58a49b346aa" alt="{\displaystyle (6-3)^{2}=8x}" aria-hidden="true" width="5863.9" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c98740c82d9e30d73bbe4063b8de96c8f1b339ff" alt="{\displaystyle 9=8x}" aria-hidden="true" width="2907.6" height="936.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b51db05bfab94ff42e8c59d033e0376f765e0e12" alt="{\displaystyle x={\frac {9}{8}}}" aria-hidden="true" width="2767.1" height="2228.5">

dengan cara bagi adil

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f8ca0c8558ac9c175e30ffbda787008499cd172" alt="{\displaystyle (y-k)(y_{1}-k)=2px+2px_{1}}" aria-hidden="true" width="12652.7" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef452865b19099e0ac66a4c22075976d179ea0e" alt="{\displaystyle (y-3)(6-3)=4x+4({\frac {9}{8}})}" aria-hidden="true" width="11772.9" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99870e3cb79712f770be5af8d9eb11e645be3298" alt="{\displaystyle (y-3)3=4x+{\frac {9}{2}}}" aria-hidden="true" width="7990.9" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1190603aa5a76c123d7fe2a454991c7139e2bf" alt="{\displaystyle 3y-9=4x+{\frac {9}{2}}}" aria-hidden="true" width="7211.9" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/530e005bd88cc5440d7617d0b12901d4d0309381" alt="{\displaystyle 3y=4x+{\frac {27}{2}}}" aria-hidden="true" width="5989" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adabfc49ee9ef63d51dbec5136b036f79e1764e6" alt="{\displaystyle y={\frac {4}{3}}x+{\frac {27}{6}}}" aria-hidden="true" width="5848.5" height="2228.5">

Tentukan persamaan garis singgung yang melalui (1,6) terhadap $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2aa603dea12dddc98204dcecf8ec862b5d2d2b" alt="{\displaystyle y^{2}-6y-8x+9=0}" aria-hidden="true" width="9028.4" height="1295.7">$ !

ubah ke bentuk sederhana

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2aa603dea12dddc98204dcecf8ec862b5d2d2b" alt="{\displaystyle y^{2}-6y-8x+9=0}" aria-hidden="true" width="9028.4" height="1295.7">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee0e75cb0abf8ce78674ce26d29dd94465b45e3" alt="{\displaystyle y^{2}-6y+9=8x}" aria-hidden="true" width="7305" height="1295.7">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240df7e655e1e9a1d926e2b24d89a00e49c3cc17" alt="{\displaystyle (y-3)^{2}=8x}" aria-hidden="true" width="5860.9" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f5050406d8021adc57e541e130c90762a6593d" alt="{\displaystyle (y-3)^{2}-8x=0makamasukkanlah(1,6)(6-3)^{2}-8(1)=9-8=1&gt;0}" aria-hidden="true" width="31982.9" height="1367.4">

(luar)

dengan cara bagi adil

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f8ca0c8558ac9c175e30ffbda787008499cd172" alt="{\displaystyle (y-k)(y_{1}-k)=2px+2px_{1}}" aria-hidden="true" width="12652.7" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faac6cdcdc7267657f082bccb687826dd04fde2b" alt="{\displaystyle (y-3)(6-3)=4x+4(1)}" aria-hidden="true" width="11412.9" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8f623fdecb6b0db638e615545d394d836d1477" alt="{\displaystyle (y-3)3=4x+4}" aria-hidden="true" width="7630.9" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66108e6df3b0c2d2045f6d9035c152a9f2a94a4e" alt="{\displaystyle 3y-9=4x+4}" aria-hidden="true" width="6851.9" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad24a31147c9c69495988accfbd103715e1f0eb2" alt="{\displaystyle 3y=4x+13}" aria-hidden="true" width="5629" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12dbb8ba28c2bfc1ae58691fc6a403124e728de6" alt="{\displaystyle y={\frac {4}{3}}x+{\frac {13}{3}}}" aria-hidden="true" width="5848.5" height="2228.5">

masukkan lah $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240df7e655e1e9a1d926e2b24d89a00e49c3cc17" alt="{\displaystyle (y-3)^{2}=8x}" aria-hidden="true" width="5860.9" height="1367.4">$

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1af6b788a9eedcff75a0042138135c832d41301" alt="{\displaystyle ({\frac {4}{3}}x+{\frac {13}{3}}-3)^{2}=8x}" aria-hidden="true" width="9380.4" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92a7d1b3fa47d1053e54bfa2cda3a4d911f2214" alt="{\displaystyle ({\frac {4}{3}}x+{\frac {4}{3}})^{2}=8x}" aria-hidden="true" width="7156.4" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/336688a17fb8cbe08af6aac44c2f54d97bdf8739" alt="{\displaystyle {\frac {16}{9}}x^{2}+{\frac {32}{9}}x+{\frac {16}{9}}-8x=0}" aria-hidden="true" width="12258.3" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b99ad4898473004626f348e338874473676ad0fa" alt="{\displaystyle {\frac {16}{9}}x^{2}-{\frac {40}{9}}x+{\frac {16}{9}}=0}" aria-hidden="true" width="9962.4" height="2228.5">

(dibagi 8/9)

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5b99cf574f108b175eb91974a5b91be39cd8fa" alt="{\displaystyle 2x^{2}+5x+2=0}" aria-hidden="true" width="7380.9" height="1223.9">

maka kita mencari nilai x

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c22777378f9c594c71158fea8946f2495f2a28" alt="{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}" aria-hidden="true" width="9267.9" height="2659.1">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8321ba86c96fa576137931183ec7986900c24ecb" alt="{\displaystyle x={\frac {5\pm {\sqrt {25-16}}}{4}}}" aria-hidden="true" width="8048.4" height="2515.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b028dca1d076bfafafb49b91b79274144b444863" alt="{\displaystyle x={\frac {5\pm {\sqrt {9}}}{4}}}" aria-hidden="true" width="5324" height="2515.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/283109827d60bf45dbab48e93e2aad8348ced115" alt="{\displaystyle x_{1}={\frac {5+{\sqrt {9}}}{4}}=2}" aria-hidden="true" width="7612.5" height="2515.6">

atau

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02ea9596400132d833a0fc0457bbec37a8b0a5c8" alt="{\displaystyle x_{2}={\frac {5-{\sqrt {9}}}{4}}={\frac {1}{2}}}" aria-hidden="true" width="7972.5" height="2515.6">

maka kita mencari nilai y

untuk

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" alt="{\displaystyle x_{1}}" aria-hidden="true" width="1026.4" height="865.1">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e42d1433b26ca18913e5ad4164720bcbf610c55" alt="{\displaystyle y_{1}={\frac {4}{3}}(2)+{\frac {13}{3}}={\frac {8}{3}}+{\frac {13}{3}}=7}" aria-hidden="true" width="13615.5" height="2228.5">

jadi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d98ec22e597396059df5bac8fcc1f2b01d97b7" alt="{\displaystyle (2,7)}" aria-hidden="true" width="2225.2" height="1223.9">$

untuk

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" alt="{\displaystyle x_{2}}" aria-hidden="true" width="1026.4" height="865.1">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aefd6621d25e8c5a514cde0de054dd44f35c05f1" alt="{\displaystyle y_{2}={\frac {4}{3}}({\frac {1}{2}})+{\frac {13}{3}}={\frac {2}{3}}+{\frac {13}{3}}=5}" aria-hidden="true" width="13975.5" height="2228.5">

jadi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1ce54eb2ee2b2759f1df54261153edfa5bc5ee" alt="{\displaystyle ({\frac {1}{2}},5)}" aria-hidden="true" width="2585.2" height="2228.5">$

kembali dengan cara bagi adil

untuk persamaan singgung pertama

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f8ca0c8558ac9c175e30ffbda787008499cd172" alt="{\displaystyle (y-k)(y_{1}-k)=2px+2px_{1}}" aria-hidden="true" width="12652.7" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13dfd6c8f6cccf80e2cd0a8e2ea0207bd9de4145" alt="{\displaystyle (y-3)(7-3)=4x+4(2)}" aria-hidden="true" width="11412.9" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea8ec966d063abcb9f237a2271e95281d7c7d53" alt="{\displaystyle (y-3)4=4x+8}" aria-hidden="true" width="7630.9" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf222f900b2aa1f82edb9745e0b168a7a118f44d" alt="{\displaystyle 4y-12=4x+8}" aria-hidden="true" width="7352.4" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c790204cf4d3eaba486e5452abed81de0290215" alt="{\displaystyle 4y=4x+20}" aria-hidden="true" width="5629" height="1080.4">

(dibagi 4)

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd098373ac90877f4dd1fb6753ffbe1728ce73b" alt="{\displaystyle y=x+5}" aria-hidden="true" width="4127.5" height="1080.4">

untuk persamaan singgung kedua

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9444bbf740f46b996ecc2c2fd51e84f8a7e6251f" alt="{\displaystyle (y-k)(y_{2}-k)=2px+2px_{2}}" aria-hidden="true" width="12652.7" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4054a967482f7c6cd5000acb40cad1b8dbe16a7a" alt="{\displaystyle (y-3)(5-3)=4x+4({\frac {1}{2}})}" aria-hidden="true" width="11772.9" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bc94d92db132acaf9c997635375bd39594fa46" alt="{\displaystyle (y-3)2=4x+2}" aria-hidden="true" width="7630.9" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee58f3e5444ca331b04a448387ad162b3c0fe4b" alt="{\displaystyle 2y-6=4x+2}" aria-hidden="true" width="6851.9" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6390cb815910b106385959c9d492f1d0d9cfcac5" alt="{\displaystyle 2y=4x+8}" aria-hidden="true" width="5128.5" height="1080.4">

(dibagi 2)

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ce95b8668dabacc3dfa077b4e83350da3357cf" alt="{\displaystyle y=2x+4}" aria-hidden="true" width="4628" height="1080.4">

Bacaan Lainnya

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Sumber bacaan: Math is Fun, Wolfram, Lumen Learning

Irisan Kerucut

3 Jenis kurva pada irisan kerucut

Geometri

Jenis-jenis irisan kerucut

Bagaimana kasus degenerasi dapat terjadi?

Geometri analitis

Koordinat Kartesius

Jika terdapat persamaan dengan bentuk:

maka:

Bentuk persamaan umum irisan kerucut

Bentuk persamaan umum irisan kerucut sebagai berikut:

kesimpulan:

Rumus irisan kerucut tentang: linkaran, parabola, elips dan hiperbola

Lingkaran

Parabola

Elips

Hiperbola

Persamaan garis singgung

dengan cara bagi adil

Contoh Soal dan Jawaban Irisan Kerucut (Cone)

Titik pusat (0,0). Tentukan persamaan garis singgung yang bergradien 2 terhadap <img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">!

Tentukan persamaan garis singgung yang melalui (4,8) terhadap <img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">!

Tentukan persamaan garis singgung yang melalui (1,5) terhadap <img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">!

jawab:

dengan cara bagi adil

masukkan lah <img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">

maka kita mencari nilai x

maka kita mencari nilai y

kembali dengan cara bagi adil

Tentukan persamaan garis singgung <img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2aa603dea12dddc98204dcecf8ec862b5d2d2b" alt="{\displaystyle y^{2}-6y-8x+9=0}" aria-hidden="true" width="9028.4" height="1295.7"> yang berordinat 6!

jawab: ubah ke bentuk sederhana

cari absis dimana ordinat 6

dengan cara bagi adil

Tentukan persamaan garis singgung yang melalui (1,6) terhadap <img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2aa603dea12dddc98204dcecf8ec862b5d2d2b" alt="{\displaystyle y^{2}-6y-8x+9=0}" aria-hidden="true" width="9028.4" height="1295.7">!

ubah ke bentuk sederhana

dengan cara bagi adil

masukkan lah <img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240df7e655e1e9a1d926e2b24d89a00e49c3cc17" alt="{\displaystyle (y-3)^{2}=8x}" aria-hidden="true" width="5860.9" height="1367.4">

maka kita mencari nilai x

maka kita mencari nilai y

jadi <img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d98ec22e597396059df5bac8fcc1f2b01d97b7" alt="{\displaystyle (2,7)}" aria-hidden="true" width="2225.2" height="1223.9">

kembali dengan cara bagi adil

Bacaan Lainnya

Unduh / Download Aplikasi HP Pinter Pandai

Titik pusat (0,0). Tentukan persamaan garis singgung yang bergradien 2 terhadap $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">$ !

Tentukan persamaan garis singgung yang melalui (4,8) terhadap $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">$ !

Tentukan persamaan garis singgung yang melalui (1,5) terhadap $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">$ !

masukkan lah $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd68256a2d490ee5e0cb1e884be43c990b1a0f23" alt="{\displaystyle y^{2}=16x}" aria-hidden="true" width="3861.1" height="1295.7">$

Tentukan persamaan garis singgung $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2aa603dea12dddc98204dcecf8ec862b5d2d2b" alt="{\displaystyle y^{2}-6y-8x+9=0}" aria-hidden="true" width="9028.4" height="1295.7">$ yang berordinat 6!

Tentukan persamaan garis singgung yang melalui (1,6) terhadap $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2aa603dea12dddc98204dcecf8ec862b5d2d2b" alt="{\displaystyle y^{2}-6y-8x+9=0}" aria-hidden="true" width="9028.4" height="1295.7">$ !

masukkan lah $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240df7e655e1e9a1d926e2b24d89a00e49c3cc17" alt="{\displaystyle (y-3)^{2}=8x}" aria-hidden="true" width="5860.9" height="1367.4">$

jadi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d98ec22e597396059df5bac8fcc1f2b01d97b7" alt="{\displaystyle (2,7)}" aria-hidden="true" width="2225.2" height="1223.9">$