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# Find the equation of the hyperbola if centre: $(0 , 0 )$ length of the semi-transverse axis is $6; e=3$ and the transverse axis is parallel to $y$-axis.

• Standard forms of equation of the hyperbola with transverse axis $2a$,conjugate axis $2t$ with the negative sign associated with $b$ and $e=\sqrt{1+\large\frac{b^2}{a^2}},b=a\sqrt{e^2-1}$
• $\large\frac{y^2}{a^2}-\frac{x^2}{b^2}$$=1 • http://clay6.com/mpaimg/2_toolbox10.jpg • Foci (0,\pm ae),centre (0,0),vertices (0,\pm a). • Transverse axis y-axis (x=0) • Conjugate axis x-axis (y=0) • End points of latus rectum (\pm\large\frac{b^2}{a},$$ae),(\pm\large\frac{b^2}{a},$$-ae) • Length of LR :\large\frac{2b^2}{a} • Directrices y=\pm\large\frac{a}{e} Step 1: C(0,0),a=6,e=3,transverse axis is parallel to y-axis. Equation is of the form \large\frac{y^2}{a^2}-\frac{x^2}{b^2}$$=1$
$b^2=a^2(e^2=1)=36(9-1)=288$
The equation is $\large\frac{y^2}{36}-\frac{x^2}{288}$$=1$