Question

The angle between the tangents to the circle with centre $(4,5)$ drawn from $P(-2,-3)$ is ${120}^o$ , the length of the tangent to the circle from $P$ is :

• A. $4$
• B. $3$
• C. $2$
• D. $5$

Answer 1

The correct option is D $5$

Given

(i) centre ${}^{\prime }{O}^{\prime }(4,5)P(-2,-3)$

(ii) $\angle BPA={120}^0$

$\Rightarrow \angle APO=\frac{1}{2}\angle APB=\frac{1}{2}\left(120\right)={60}^0$

$DP=\sqrt{(x_2-1x_1{)}^2+(y_2-y_1{)}^2}=\sqrt{(4+2{)}^2+(5+3{)}^2}$

$OP=10$

In triangle $APO$

$\cos \angle APO=\frac{AP}{PO}$

$\cos 60=\frac{AP}{10}$

$\frac{1}{2}=\frac{AP}{10}$

$\Rightarrow AP=5unit$

$\therefore$ length of tangent is $5unit$