Question

The length of diameter of director circle of hyperbola $\frac{x^2}{49}-\frac{y^2}{25}=1$, is

• A. $4$
• B. $6$
• C. $4\sqrt{6}$
• D. $24$

Answer 1

Answer: (C)

$4\sqrt{6}$

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 $x^2+y^2=a^2-b^2$

Here the given hyperbola is $\frac{x^2}{49}-\frac{y^2}{25}=1$,

Here $a=7$ and $b=5$

So equation of the director circle will be $x^2+y^2=(7{)}^2-(5{)}^2$

$\Rightarrow x^2+y^2=24$

Hence the radius of the director circle is $\sqrt{24}=2\sqrt{6}$. So the diameter will be $4\sqrt{6}$.

So correct option is $C$.