Question

The locus of the point, whose chord of contact w.r.t the circle $x^2+y^2=a^2$ makes an angle $2\alpha$ at the centre of the circle is

• A. $x^2+y^2=2a^2$
• B. $x^2+y^2=a^{2}co{s}^2\alpha$
• C. $x^2+y^2=a^{2}se{c}^2\alpha$
• D. $x^2+y^2=a^{2}ta{n}^2\alpha$

Answer 1

The correct option is C $x^2+y^2=a^{2}se{c}^2\alpha$


Let $P(x_1,y_1)$ be a point on the locus
Let PA and PB be the tangent drawn from P to the circle $x^2+y^2=a^2$
And the chord AB makes an angle $2\alpha$ at the origin O(0, 0) the centre of the circle.
Then $\angle AOP=\alpha ,cos\alpha =\frac{OA}{OP}\Rightarrow sec\alpha =\frac{OP}{OA}$
$\Rightarrow O{P}^2=O{A}^{2}se{c}^2\alpha \Rightarrow x_1^2+y_1^2=a^{2}se{c}^2\alpha \therefore Locusof(x_1,y_1)is{x}^2+y^2=a^{2}se{c}^2\alpha$