Question

A line is drawn through the point $A(3,4)$ inclined at an acute angle $\theta$ with the $+ive$ direction of $x-$axis. If it cuts the circle $x^2+y^2=4$ in $B$ and $C$ then the lengths $AB$ and $AC$ are the roots of
$r^2+2(3\cos \theta +4\sin \theta )r+21=0$.

• A. True
• B. False

Answer 1

The correct option is A True

$\frac{x-3}{\cos \theta }=\frac{y-4}{\sin \theta }=r=AB$ or $AC$ where $A$ is $(3,4)$
${(r\cos \theta +3)}^2+{(r\sin \theta +4)}^2=4$.

$r^2+2(3\cos \theta +4\sin \theta )r+21=0$.
This is same as given quadratic in $r$ which gives $AB$ and $AC$ the two values of $r$.