Soal-Soal Matematika/Persamaan elips

Persamaan elips

	Vertikal	Horisontal
	Titik pusat (0,0)
Persamaan	${\frac {x^{2}}{b^{2}}}+{\frac {y^{2}}{a^{2}}}=1$	${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$
Panjang sumbu mayor	$2a$	$2a$
Panjang sumbu minor	$2b$	$2b$
Panjang Latus Rectum	$L={\frac {2b^{2}}{a}}$	$L={\frac {2b^{2}}{a}}$
Fokus	$F(0,\pm c)$	$F(\pm c,0)$
Puncak	$P(0,\pm a)$	$P(\pm a,0)$
Direktris	$y=\pm {\frac {a^{2}}{c}}$	$x=\pm {\frac {a^{2}}{c}}$
Eksentrisitas	$e={\frac {c}{a}}$	$e={\frac {c}{a}}$
	Titik pusat (h,k)
Persamaan	${\frac {(x-h)^{2}}{b^{2}}}+{\frac {(y-k)^{2}}{a^{2}}}=1$	${\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1$
Panjang sumbu mayor	$2a$	$2a$
Panjang sumbu minor	$2b$	$2b$
Panjang Latus Rectum	$L={\frac {2b^{2}}{a}}$	$L={\frac {2b^{2}}{a}}$
Fokus	$F(h,k\pm c)$	$F(h\pm c,k)$
Puncak	$P(h,k\pm a)$	$P(h\pm a,k)$
Direktris	$y=\pm {\frac {a^{2}}{c}}$	$x=\pm {\frac {a^{2}}{c}}$
Eksentrisitas	$e={\frac {c}{a}}$	$e={\frac {c}{a}}$

dimana $c={\sqrt {a^{2}-b^{2}}}$

Persamaan garis singgung

bergradien $m$ ($y=mx+c$)

Vertikal	Horisontal
Titik pusat (0,0)
$y=mx\pm {\sqrt {b^{2}+a^{2}m^{2}}}$	$y=mx\pm {\sqrt {a^{2}m^{2}+b^{2}}}$
Titik pusat (h,k)
$(y-k)=m(x-h)\pm {\sqrt {b^{2}+a^{2}m^{2}}}$	$(y-k)=m(x-h)\pm {\sqrt {a^{2}m^{2}+b^{2}}}$

jika persamaan garis lurus bergradien sejajar maka $m_{2}=m_{1}$

jika persamaan garis lurus bergradien tegak lurus maka $m_{2}={\frac {-1}{m_{1}}}$

melalui titik $(x_{1},y_{1})$

dengan cara bagi adil

Vertikal	Horisontal
Titik pusat (0,0)
${\frac {xx_{1}}{b^{2}}}+{\frac {yy_{1}}{a^{2}}}=1$	${\frac {xx_{1}}{a^{2}}}+{\frac {yy_{1}}{b^{2}}}=1$
Titik pusat (h,k)
${\frac {(x-h)(x_{1}-h)}{b^{2}}}+{\frac {(y-k)(y_{1}-k)}{a^{2}}}=1$	${\frac {(x-h)(x_{1}-h)}{a^{2}}}+{\frac {(y-k)(y_{1}-k)}{b^{2}}}=1$

jika titik $(x_{1},y_{1})$ berada di dalam bentuknya maka ada 1 persamaan garis singgung (1 langkah).

jika titik $(x_{1},y_{1})$ berada di luar bentuknya maka ada 2 persamaan garis singgung (2 langkah).