Question

A hyperbola intersects an ellipse $x^2+9y^2=9$ orthogonally. The eccentricity of the hyperbola is reciprocal of that of ellipse. If the axes of the hyperbola are along coordinate axes, then

• A. vertices of hyperbola are $(\pm \frac{8}{3},0)$
• B. y coordinate of point of intersection of ellipse and hyperbola is either $\frac{1}{3}or-\frac{1}{3}$
• C. latus rectum of hyperbola is $\frac{2}{3}$
• D. latus rectum of hyperbola is $\frac{4}{3}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A

vertices of hyperbola are $(\pm \frac{8}{3},0)$

B

y coordinate of point of intersection of ellipse and hyperbola is either $\frac{1}{3}or-\frac{1}{3}$

C

latus rectum of hyperbola is $\frac{2}{3}$

Hyperbola and ellipse will be confocal with focus $(\pm 2\sqrt{2},0)$

$1-e_1^2=\frac{1}{9}\Rightarrow e_1=\frac{2\sqrt{2}}{3}\left(ellipse\right)e_2=\frac{3}{2\sqrt{2}}\left(hyperbola\right)\frac{x^2}{A^2}-\frac{y^2}{B^2}=1\Rightarrow A=\frac{8}{3}{B}^2=\frac{64}{9}(\frac{9}{8}-1)\Rightarrow B^1=\frac{8}{9}\therefore \frac{9x^2}{64}-\frac{9y^2}{8}=1\left(equationofhyperbola\right)$