Question

If the line $x\cos \alpha +y\sin \alpha =p$ intersects the circle $x^2+y^2=4$ at $A$ and $B$ and chord $AB$ makes an angle of ${30}^{\circ }$ at a point on the circumference of circle then $3p^2=$

Answer 1

$\angle AOB=2\angle AXB={60}^{\circ }$
$\because p$ is perpendicualar distance of the given line from origin
$\therefore p=OD$
Now In $△OAB$
$\cos 30°=\frac{p}{2}$
$\Rightarrow p=\sqrt{3}$
$\Rightarrow 3p^2=9$