Question

The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points $(4,-1)$ and $(-2,2)$ is

• A. $\frac{1}{2}$
• B. $\frac{2}{\sqrt{5}}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\frac{\sqrt{3}}{4}$

Answer 1

The correct option is C $\frac{\sqrt{3}}{2}$

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

Let $\frac{1}{a^2}=A,\frac{1}{b^2}=B$

$A{x}^2+B{y}^2=1$

$16A+B=1$ $(4,-1)$ .....$\left(1\right)$

$4A+4B=1$ $(-2,2)$

$A+B=\frac{1}{4}$ .....$\left(2\right)$

Subtracting $\left(2\right)$ from $\left(1\right)$,

$15A=\frac{3}{4}$

$\Rightarrow A=\frac{1}{20}$ $\Rightarrow a^2=20$

and $B=1-\frac{16}{20}$ $=\frac{4}{20}=\frac{1}{5}$

$\Rightarrow b^2=5$

$b^2=a^2(1-e^2)$

$5=20(1-e^2)$

$\frac{1}{4}=1-e^2$

$e^2=\frac{3}{4}$

$e=\frac{\sqrt{3}}{2}$

Hence, option $C$ is correct.