Find the coordinates of the foci, the vertices, the length of majo | Filo

Class 11

Math

3D Geometry

Conic Sections

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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

Solution: The given equation is $$\displaystyle 4x^{2}+9y^{2}= 36$$
It can be written as
$$\displaystyle 4x^{2}+9y^{2}=36$$
$$\displaystyle \frac{x^{2}}{9}+\frac{y^{2}}{4}= 1$$
$$\displaystyle \frac{x^{2}}{3^{2}}+\frac{y^{2}}{2^{2}}= 1 ...(1)$$
Here the denominator of $$\displaystyle \frac{x^{2}}{3^{2}}$$ is greater than the denominator of $$\displaystyle \frac{y^{2}}{2^{2}}$$
Therefore, the major axis is along the $$x$$-axis while the minor axis is along the $$y$$-axis.
On comparing, the given equation with $$\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}= 1$$, we obtain $$a = 3$$ and $$b = 2$$
$$\displaystyle \therefore ae= c =\sqrt{a^{2}-b^{2}}= \sqrt{9 - 4}= \sqrt{5}$$
Therefore, the coordinates of the foci are $$\displaystyle \left ( \pm \sqrt{5}, 0 \right )$$.
The coordinates of the vertices are $$\displaystyle \left ( \pm 3, 0 \right )$$
Length of major axis $$= 2a = 6$$
Length of minor axis $$= 2b = 4$$
Eccentricity $$e =$$ $$\displaystyle \frac{c}{a}=\frac{\sqrt{5}}{3}$$
Length of latus rectum $$=$$ $$\displaystyle \frac{2b^{2}}{a}= \frac{2\times 4}{3}=\frac{8}{3}$$
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