Question

The line y = x intersects the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{25} = 1$ at the points P and Q. The eccentricity of ellipse with PQ as majoraxis and minor axis of length $\frac{5}{\sqrt{2}}$ is

• A. $\frac{\sqrt{5}}{3}$
• B. $\frac{5}{\sqrt{3}}$
• C. $\frac{5}{9}$
• D. $\frac{2 \sqrt{2}}{9}$

Answer 1

Answer: (A)

Given equation of hyperbola and line are
$\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$ and $y=x$ respectively.
For intersection point of both curve put $y=x$, we get
$ \frac{x^{2}}{9}-\frac{x^{2}}{25} =1 $
$\Rightarrow x^{2} =\frac{9 \times 25}{16}=\left(\frac{15}{4}\right)^{2}$
$\Rightarrow x =\pm \frac{15}{4} \text { and } y=\pm \frac{15}{4}$
$\therefore $ Intersetion points $P\left(\frac{15}{4}, \frac{15}{4}\right)$
and
$Q\left(\frac{-15}{4}, \frac{-15}{4}\right)$
Since, $P Q$ is major axis, then its length
$=2 \sqrt{2} \cdot \frac{15}{4}=\frac{15}{\sqrt{2}}$
and length of minor axis is $\frac{5}{\sqrt{2}}$ (given)
i.e., Major axis, $2 a=\frac{15}{\sqrt{2}} $
$\Rightarrow a=\frac{15}{2 \sqrt{2}}$
and minor axis, $2 b=\frac{5}{\sqrt{2}} $
$\Rightarrow b=\frac{5}{2 \sqrt{2}}$
$\therefore $ Eccentricity of an ellipse
$=\sqrt{\frac{a^{2}-b^{2}}{a^{2}}}=\sqrt{1-\left(\frac{b}{a}\right)^{2}} $
$=\sqrt{1-\left(\frac{1}{3}\right)^{2}}=\sqrt{\frac{8}{9}}=\frac{2 \sqrt{2}}{9}$