Question

If the line joining the foci of the hyperbola $S_{1}\equiv \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}+1=0$ does not subtend a right angle at any point on the hyperbola $S_{2}\equiv \frac{x^{2}}{4 a^{2}}-\frac{y^{2}}{ b^{2}}=1,$ then number of integral values of $4e^{2}$ is/are ( $e$ is eccentricity of $S_{2}=0$ )

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (B)

Circle having foci of the hyperbola $S_{1}$ as the extremities of diameter should not intersect the hyperbola $S_{2}$ at real points
$\Rightarrow a^{2}+b^{2} < 4a^{2}$
$\Rightarrow b^{2} < 3a^{2}$
$\Rightarrow e^{2} < 7/4$
$\Rightarrow 4 < 4e^{2} < 7$