Find the coordinates of the foci, the vertices, the length of major axis, the minor axis,

Question:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $4 x^{2}+9 y^{2}=36$

Solution:

The given equation is $4 x^{2}+9 y^{2}=36$.

It can be written as

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

Or, $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$

$\mathrm{Or}, \frac{x^{2}}{3^{2}}+\frac{y^{2}}{2^{2}}=1$ $\ldots(1)$

Here, the denominator of $\frac{x^{2}}{3^{2}}$ is greater than the denominator of $\frac{y^{2}}{2^{2}}$.

Therefore, the major axis is along the $x$-axis, while the minor axis is along the $y$-axis.

On comparing the given equation with $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, we obtain $a=3$ and $b=2$.

$\therefore c=\sqrt{a^{2}-b^{2}}=\sqrt{9-4}=\sqrt{5}$

Therefore,

The coordinates of the foci are $(\pm \sqrt{5}, 0)$.

The coordinates of the vertices are (±3, 0).

Length of major axis = 2a = 6

Length of minor axis = 2b = 4

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

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

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