Question

From any point P on the circle $x^2+y^2=9$ two tangents are drawn to circle $x^2+y^2=4$ which cut $x^2+y^2=9$ at A and B then locus of point of intersection of tangents drawn $x^2+y^2=9$ at A and B is the curve

• A. $x^2+y^2=27$
• B. $x^2+y^2={27}^2$
• C. $x^2+2y^2={27}^2$
• D. $x^2+y^2=8$

Answer 1

Answer: (B)

$x^2+y^2={27}^2$

We have $\sin \theta =\frac{2}{3}\Rightarrow \tan \theta =\frac{2}{\sqrt{5}}$

$\cos 2\theta =\frac{3}{OR}$

$\Rightarrow \frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }=\frac{3}{OR}\Rightarrow \frac{1-\frac{4}{5}}{1+\frac{4}{5}}=\frac{3}{OR}$
$\Rightarrow OR=27$

$\Rightarrow$R is situated at a fixed distance from O

$\Rightarrow$ Locus $x^2+y^2={27}^2$