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, $PQ$ 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, $2a=\frac{15}{\sqrt{2}}$
$\Rightarrow a=\frac{15}{2\sqrt{2}}$
and minor axis, $2b=\frac{5}{\sqrt{2}}$
$\Rightarrow b=\frac{5}{2\sqrt{2}}$
$\therefore$ Eccentricity of an ellipse
$=\sqrt{\frac{a^2-b^2}{a^2}}=\sqrt{1-{\left(\frac{b}{a}\right)}^2}$
$=\sqrt{1-{\left(\frac{1}{3}\right)}^2}=\sqrt{\frac{8}{9}}=\frac{2\sqrt{2}}{9}$