Question

If angle between two tangents drawn from a point $P$ to a circle of radius $a$ and center $O$ is $90°$, then $OP=a\sqrt{2}$. Write ‘True’ or ‘False’ and justify your answer.

• A. True
• B. False

Answer 1

The correct option is A True

Verify the given statement

Let $PT$ and $PR$ be the tangents drawn from point $P$ to the circle.

According to given condition $\angle TPR={90}^{\circ }$

The line joining the centre and the point $P$ is an angle bisector of the $\angle TPR$

So, $\angle TPO=\angle OPR={45}^{\circ }$

We know that the tangent to a circle is perpendicular to the radius at the point of contact.

So, $OT\perp TP$

In right angled $△TPO$

$\sin {45}^{\circ }=\frac{OT}{OP}$

$\Rightarrow$ $\frac{1}{\sqrt{2}}=\frac{a}{OP}$

$\Rightarrow$ $OP=a\sqrt{2}$

Hence the given statement is true.