Question

If a line id drawn through a point $A(3,4)$ to cut the circle $x^2+y^2=4$ at $P$ and $Q$ then $AP.AQ=$

• A. $15$
• B. $17$
• C. $21$
• D. $25$

Answer 1

The correct option is A $15$

Given circle $x^2+y^2=4$ cuts at $P$ and $Q$

Centre is $C(0,0)$ and radius $=r=\sqrt{4}=2$ units.

Power of circle$=AP.AQ={AT}^2$ .......$\left(1\right)$

In $△ACT,\angle T={90}^{\circ }$

${CT}^2+{TA}^2={CA}^2$

$\Rightarrow r^2+{TA}^2={\left(\sqrt{{(3-0)}^2+{(4-0)}^2}\right)}^2$

$\Rightarrow 4+{TA}^2=9+16=25$

$\Rightarrow {TA}^2=25-4=21$

$\therefore AP.AQ=21$