Question

To draw a pair of tangents to a circle which are inclined to each other at an angle of ${75}^{\circ }$. It is required to draw tangents at the endpoints of two radii of the circle, the angle between them should be.

• A. ${105}^{\circ }$
• B. ${65}^{\circ }$
• C. ${95}^{\circ }$
• D. ${75}^{\circ }$

Answer 1

The correct option is A ${105}^{\circ }$

PA and PB are tangents drawn from an external point P to the circle.

$\angle OAP=\angle OBP={90}^{\circ }$ (Radius is perpendicular to the tangent at point of contact)

In quadrilateral OAPB,

$\angle APB+\angle OAD+\angle AOD+\angle OBP={360}^{\circ }$

therefore ${75}^{\circ }+{90}^{\circ }+\angle AOB+{90}^{\circ }$ =${360}^{\circ }$

= ${255}^{\circ }+\angle AOB$= ${360}^{\circ }$

=$\angle AOB={360}^{\circ }–{255}^{\circ }={105}^{\circ }$

Thus, the angle between the two radius, OA and OB is ${105}^{\circ }$

therefore option A is the answer.