Question

The radius of the director circle of hyperbola $$\dfrac{\mathrm x^2}{\mathrm a^2}-\dfrac{\mathrm y^2}{\mathrm b^2}=1$$ is
A. a
B. b
$$\mathrm C.\;\sqrt{\mathrm a^2+\mathrm b^2}\\\mathrm D.\;\sqrt{\mathrm a^2-\mathrm b^2}$$

Answer 1

Hint:The director circle of a hyperbola is defined as the locus of the point of intersection of two perpendicular tangents to the hyperbola. The equation of the director circle of a general hyperbola is given by-$x^2 + y^2 = a^2 - b^2$.Comparing it with the general equation of circle we get the radius of the director circle of hyperbola.

Complete step-by-step answer:
The given equation of hyperbola is a general equation. So the equation of its director circle is given by-
$x^2 + y^2 = a^2 - b^2$.....(1)

Now, we have to find the radius of this circle. The general equation of a circle at centre (0, 0) is given by $x^2 + y^2 = r^2$
, where r is the radius.
By comparing (1) and (2),
$r^2 = a^2 - b^2$
$$\mathrm r=\sqrt{\mathrm a^2-\mathrm b^2}$$
Hence the Director circle is a circle whose centre is same as centre of the hyperbola and the radius is $$\sqrt{\mathrm a^2-\mathrm b^2}$$
This is the required answer, the correct option is D.

Note: One should know the definition and formula of the director circle.Even if formula is not known, we can use the definition to find the locus of points that form a director circle.The locus of point of intersections of perpendicular tangents to the hyperbola is a circle called as Director circle and for a standard hyperbola $$\dfrac{\mathrm x^2}{\mathrm a^2}-\dfrac{\mathrm y^2}{\mathrm b^2}=1$$, its equation is $x^2+y^2 = a^2-b^2$.