First slide

Ellipse

Question

Equation of an ellipse with centre at the origin passing through $(5, 0)$ and having eccentricity $2 / 3$ is.

Moderate

A

$16 x^{2} + 5 y^{2} = 25$
B

$9 x^{2} + 5 y^{2} = 25$
C

$5 x^{2} + 9 y^{2} = 25$
D

$5 x^{2} + 16 y^{2} = 25$

Solution

Let the equation of the ellipse be $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ .
It passes through $(5, 0) \Rightarrow a^{2} = 25$
$b^{2} = a^{2} (1 - e^{2}) \Rightarrow b^{2} = 25 (1 - \frac{4}{9}) = \frac{125}{9}$
Hence the required equation of the ellipse is
$5 x^{2} + 9 y^{2} = 25$

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