Find the equation of director circle for ellipse $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$?

Question

$\begin{array}{1 1}(a)\;x^2-y^2=a^2+b^2\\(b)\;x^2+y^2=a^2+b^2\\(c)\;x^2-y^2=a^2-b^2\\(d)\;\text{None of these}\end{array}$

Answer 1

Toolbox:

• Director circle is locus of point of intersection of $\perp$ tangents to the ellipse.

http://clay6.com/mpaimg/Jeemain_q86.png

Let any tangent in terms of slope of ellipse $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$ is

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

It passes through (h,k)

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

$(k-mh)^2=a^2m^2+b^2$

$k^2+m^2h^2-2mhk=a^2m^2+b^2$

$m^2(h^2-a^2)-2hkm+k^2-b^2=0$

It is quadratic equation in m let slope of two tangent are $m_1$ and $m_2$

$m_1m_2=\large\frac{k^2-b^2}{h^2-a^2}$

$-1=\large\frac{k^2-b^2}{h^2-a^2}$(tangents are $\perp)$

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

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

This is the required equation.

Hence (b) is the correct answer.