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 major axis 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. $\frac{25}{9}$

Answer 1

The correct option is A $\frac{\sqrt{5}}{3}$

Line $y=x$ intersects hyperbola $\frac{x^2}{9}-\frac{y^2}{25}=1$ at P and Q putting $y=x$ in eq. of parabola

$\frac{x^2}{9}-\frac{y^2}{25}=1$

$16y^2=9\times 25$

$y=\pm \frac{3\times 5}{4}$

$x=\pm \frac{3\times 5}{4}$

PQ = Distance b/w points $(\frac{+15}{4},\frac{+15}{4})$ and $(\frac{-15}{4},\frac{-15}{4})$

$=\sqrt{{\left(\frac{15}{2}\right)}^2+{\left(\frac{15}{2}\right)}^2}$

$=\frac{15}{\sqrt{2}}$

Length of major axis $\Rightarrow \frac{15}{\sqrt{2}}$, length of minor axis $=\frac{5}{\sqrt{2}}$

$e=\sqrt{1-\frac{{\left(\frac{5}{\sqrt{2}}\right)}^2}{{\left(\frac{15}{\sqrt{2}}\right)}^2}}\Rightarrow e=\sqrt{\frac{200}{225}}\Rightarrow \frac{2\sqrt{2}}{3}$