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9. 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.

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

Answers (1)

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Given

The equation of the ellipse

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

$\frac{4x^2}{36} + \frac{9y^2}{36} = 1$

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

$\frac{x^2}{3^2} + \frac{y^2}{2^2} = 1$

As we can see from the equation, the major axis is along X-axis and the minor axis is along Y-axis.

On comparing the given equation with the standard equation of an ellipse, which is

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

We get

$a=3$ and $b=2$ .

So,

$c=\sqrt{a^2-b^2}=\sqrt{3^2-2^2}$

$c=\sqrt{5}$

Hence,

Coordinates of the foci:

$(c,0)\:and\:(-c,0)=(\sqrt{5},0)\:and\:(-\sqrt{5},0)$

The vertices:

$(a,0)\:and\:(-a,0)=(3,0)\:and\:(-3,0)$

The length of the major axis:

$2a=2(3)=6$

The length of minor axis:

$2b=2(2)=4$

The eccentricity :

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

The length of the latus rectum:

$\frac{2b^2}{a}=\frac{2(2)^2}{3}=\frac{8}{3}$

Posted by

Pankaj Sanodiya

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