Question

The equation of director circle of $-\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, If $b<a$ is:

• A. $x^2+y^2=b^2-a^2$
• B. $x^2+y^2=b^2+a^2$
• C. $x^2-y^2=b^2-a^2$
• D. Director circle does not exist

Answer 1

The correct option is D Director circle does not exist

The Director circle of a hyperbola is defined as the locus of the point of intersection of two perpendicular tangents to the hyperbola. For any standard hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,

The equation of Director circle is given by $x^2+y^2=a^2-b^2$

The given hyperbola $-\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is a conjugate hyperbola.

Hence the equation of Director circle for a conjugate hyperbola is given by $x^2+y^2=b^2-a^2$

Hence the Director circle of given hyperbola is a circle whose center is same as center of the given hyperbola and the radius is $\sqrt{b^2-a^2}$

As the radius is always a positive and real value, so $(b^2-a^2)>0$

$\to (b-a)(b+a)>0$

As $a$ and $b$ both are positive quantities hence $a+b>0$

Hence $b-a>0$ or $b>a$

For $b<a$ the director circle does not exist, as the radius will not be real for $b<a$

So correct option is $D$.