Question

If the tangents drawn from point $P(-4,0)$ to the circle $x^2+y^2=4$ touches the circle at points $Q$ and $R,$ then $QR=$

• A. $\sqrt{3}$
• B. $4\sqrt{3}$
• C. $2\sqrt{3}$
• D. $4$

Answer 1

The correct option is C $2\sqrt{3}$

Given, $x^2+y^2=4\Rightarrow 2x+2y\frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=-\frac{x}{y}$

Let $Q\equiv (x_1,y_1)$

$\therefore -\frac{x_1}{y_1}=\frac{y_1-0}{x_1+4}\Rightarrow y_1^2=-x_1^2-4x_1$

$\Rightarrow 4-x_1^2=-x_1^2-4x_1(\because x_1^2+y_1^2=4)$

$x_1=-1$

$y_1=\pm \sqrt{3}\Rightarrow QR=2\sqrt{3}$