Question

Find the co-ordinates of the foci, the vertices, the length of major axis, latus rectum and the eccentricity of the conic represented by the equation $9x^2+16y^2=144$

Answer 1

The equation of the given conic is $9x^2+16y^2=144$

It can be written as $\frac{x^2}{16}+\frac{y^2}{9}=1$ ........$\left(1\right)$

which is comparable with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

So $\left(1\right)$ represents an ellipse

Here $a^2=16,b^2=9\Rightarrow a=4,b=3$

We know that $c=\sqrt{a^2-b^2}=\sqrt{16-9}=\sqrt{7}$

$\therefore$ the coordinates of foci are $(-c,0)$ and $(c,0)$

i.e.,$(-\sqrt{7},0)$ and $(\sqrt{7},0)$

The co-ordinate of vertices are $(-a,0)$ and $(a,0)$

i.e.,$(-4,0)$ and $(4,0)$

Length of major axis$=2a=2\times 4=8$

Length of minor axis$=2b=2\times 3=6$

Length of latus rectum$=\frac{2b^2}{a}=\frac{2\times 3^2}{4}=\frac{2\times 9}{4}=\frac{9}{2}$

Eccentricity$=\frac{c}{a}=\frac{\sqrt{7}}{2}$