Question

With one focus of the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1$ as the centre, a circle is drawn which is tangent to the hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is :

• A. Less than or equal to $2$
• B. $2$
• C. $\frac{11}{3}$
• D. None of these

Answer 1

The correct option is A Less than or equal to $2$

Equation of hyperbola ,$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\frac{x^2}{3^2}-\frac{y^2}{4^2}=1$

So, $a=3,b=4$

focus of the hyperbola $=(c,0)$

$c=\sqrt{a^2+b^2}$

$c=\sqrt{3^2+4^2}$

$c=\sqrt{25}$

$c=5$

Maximum radius$=c-a$

$=5-3$

$=2$

Tangent to the circle can any point on the hyperbola

So, the value of radius is less than or equal to $2$