Question

The angle between pair of tangents drawn from any point on the circle $x^2+y^2=a^2$ upon the circle $x^2+y^2=b^2$ is $\frac{\pi }{3}$. Then, the locus of mid point of chord of contact is

• A. $x^2+y^2=\frac{5a^2-b^2}{4}$
• B. $x^2+y^2=\frac{5b^2-a^2}{4}$
• C. $x^2+y^2=\frac{3a^2-b^2}{4}$
• D. $x^2+y^2=\frac{4a^2-b^2}{4}$

Answer 1

The correct option is B $x^2+y^2=\frac{5b^2-a^2}{4}$


Length of tangent from $(x_1,y_1)$ to $x^2+y^2=b^2$
is $\sqrt{x_1^2+y_1^2-b^2}$ i.e $\sqrt{a^2-b^2}$.
Let $(\alpha ,\beta )$ is mid point of chord of contact.
In $△PQA,\sin {30}^{\circ }=\frac{QA}{PA}$
$\Rightarrow QA=\frac{\sqrt{a^2-b^2}}{2}$
But $QA$ is semi length of chord of contact.
$\therefore \sqrt{b^2-({\alpha }^2+{\beta }^2)}=\frac{\sqrt{a^2-b^2}}{2}$
$\Rightarrow b^2-\frac{a^2-b^2}{4}={\alpha }^2+{\beta }^2$
$\therefore$ The locus is $x^2+y^2=\frac{5b^2-a^2}{4}$