Question

A tangent $PQ$ at a point $P$ of a circle of radius $5cm$ meets a line through the centre $O$ at a point $Q$ so that $OQ=12cm$. Length $PQ$ is :

• A. $12cm$
• B. $13cm$
• C. $8.5cm$
• D. $\sqrt{119}cm$

Answer 1

The correct option is D

$\sqrt{119}cm$

Step 1. From the figure, the line that is drawn from the centre of the given circle to the tangent$PQ$ is perpendicular to $PQ$

So, $OP\perp PQ$

Step 2. Using Pythagoras theorem in right angled triangle $∆OPQ$ we have,

$$\begin{gathered} O{Q}^2=O{P}^2+P{Q}^2 \\ \Rightarrow {12}^2=5^2+P{Q}^2 \\ \Rightarrow 144=25+P{Q}^2 \\ \Rightarrow 144-25=P{Q}^2 \\ \Rightarrow 119=P{Q}^2 \\ \Rightarrow P{Q}^2=119 \\ \Rightarrow PQ=\sqrt{119} \end{gathered}$$

Hence option (D) is the correct option.