Question

The locus of the points of intersection of the perpendicular tangents drawn to the hyperbola $\frac{x^2}{12}-\frac{y^2}{8}=1$ is the curve $S=0$. Again, the locus of the points of intersection of the perpendicular tangents drawn to the curve $S=0$ is another curve $S^{\prime }=0$. Then which of the following is TRUE?

• A. $S=0$ and $S^{\prime }=0$ intersect in exactly $4$ points.
• B. Tangent to $S^{\prime }=0$ intersect $S=0$ in exactly $2$ points.
• C. Area bounded by the curve $S=0$ is $20\pi$ sq.units.
• D. Area bounded by the curve $S^{\prime }=0$ is $8\pi$ sq. units

Answer 1

The correct option is D Area bounded by the curve $S^{\prime }=0$ is $8\pi$ sq. units

Given Hyperbola: $\frac{x^2}{12}-\frac{y^2}{8}=1$
By definition, $S=0$ is the Director circle of the Hyperbola.
$\therefore S\equiv x^2+y^2-4=0(\because x^2+y^2=a^2-b^2)$
Again, by given condition, $S^{\prime }=0$ is the Director circle of $S=0$
$\therefore S^{\prime }=x^2+y^2-8=0(\because x^2+y^2=2r^2)$

$S=0$ and $S^{\prime }=0$ do not intersect

Tangent to $S^{\prime }=0$ can't intersect $S=0$ because $S=0$ lies inside $S^{\prime }=0.$

Area bounded by $S=0$ is $4\pi .$

Area bounded by $S^{\prime }=0$ is $8\pi .$