Question

To draw a pair of tangents to a circle which are inclined to each other at an angle of ${30}^{\circ }$, it is required to draw tangents at end points of those two radii of the circle, the angle between them should be

• A. ${60}^{\circ }$
• B. ${150}^{\circ }$
• C. ${90}^{\circ }$
• D. ${120}^{\circ }$

Answer 1

The correct option is B ${150}^{\circ }$

Let PA and PB are tangents to the circle with centre O, at points A and B respectively. Both tangents are inclined to each other at ${30}^{\circ }$.


$\therefore \angle APB={30}^{\circ }$ ..… (i)

We know that the radius of circle is perpendicular to its tangent.
$\therefore OA\perp PAandOB\perp PB$

Now, in quadrilateral PBOA,
$\angle APB+\angle PBO+\angle AOB+\angle OAP={360}^{\circ }$
$\Rightarrow {30}^{\circ }+{90}^{\circ }+\angle AOB+{90}^{\circ }={360}^{\circ }$
$\Rightarrow \angle AOB={360}^{\circ }-{210}^{\circ }={150}^{\circ }$
Therefore, the angle between the radii of the circle is ${150}^{\circ }$.

Hence, the correct answer is option b.