Question

The locus of a point of intersection of perpendicular tangents of the hyperbola is called

• A. auxillary circle
• B. circumcentre
• C. director circle
• D. incentre

Answer 1

The correct option is C director circle

The general equation of tangent for any standard hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ in terms of slope $m$ is given by:

$y=mx+\sqrt{a^2{m}^2-b^2}$

Let the point of intersection of two perpendicular tangents is $P(h,k)$. The equation of tangent passes through the point $P$,

hence $k=mh+\sqrt{a^2{m}^2-b^2}$

$\to (k-mh{)}^2=a^2{m}^2-b^2$

$\to m^2(h^2-a^2)-2hkm+k^2+b^2=0$ ...$\left(1\right)$

Let the slope of both tangents be $m_1$ and $m_2$.

The equation $\left(1\right)$ is a quadratic equation in $m$, where $m$ is the slope of tangents. The roots of this equations are $m_1$ and $m_2$, which are slopes of tangents perpendicular to each other.

From quadratic equation, $m_1{m}_2=\frac{k^2+b^2}{h^2-a^2}$

As both tangents are perpendicular too, hence $m_1{m}_2=-1$

By comparing both values we get,

$\to -1=\frac{k^2+b^2}{h^2-a^2}$

Hence $h^2+k^2=a^2-b^2$

Now taking point $P(h,k)$ as a variable point $P(x,y)$, we get the locus of the point of intersection of two perpendicular tangents of the hyperbola.

So locus is $x^2+y^2=a^2-b^2$

Which is exactly same as the Director circle of the hyperbola. Hence correct option is $C$.