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 $4x^2+9y^2=36$

Answer 1

The equation of the given conic is $4x^2+9y^2=36$

It can be written as $\frac{x^2}{9}+\frac{y^2}{4}=1$ ........$\left(1\right)$

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

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

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

We know that $c=\sqrt{a^2-b^2}=\sqrt{9-4}=\sqrt{5}$

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

i.e.,$(-\sqrt{5},0)$ and $(\sqrt{5},0)$

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

i.e.,$(-3,0)$ and $(3,0)$

Length of major axis$=2a=2\times 3=6$

Length of minor axis$=2b=2\times 2=4$

Length of latus rectum$=\frac{2b^2}{a}=\frac{2\times 2^2}{3}=\frac{2\times 4}{3}=\frac{8}{3}$

Eccentricity$=\frac{c}{a}=\frac{\sqrt{5}}{3}$