Question

In the given figure, PA and PB are tangents to a circle centred at O. If \( \angle OAB = 15^\circ \), then \( \angle APB \) equals :

• A. \( 30^\circ \)
• B. \( 15^\circ \)
• C. \( 45^\circ \)
• D. \( 10^\circ \)

Hint:

The angle between tangents is always double the angle between the chord and the tangent (\( \angle APB = 2 \times \angle OAB \)).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Radius is perpendicular to the tangent. Sum of angles in a triangle is \( 180^\circ \).
Step 2: Detailed Explanation:
In \( \triangle OAB \), \( OA = OB \) (radii).
So, \( \angle OBA = \angle OAB = 15^\circ \).
\( \angle AOB = 180^\circ - (15^\circ + 15^\circ) = 150^\circ \).
In quadrilateral OAPB, \( \angle OAP = \angle OBP = 90^\circ \).
Sum of angles \( \angle APB + \angle AOB = 180^\circ \) (supplementary).
\[ \angle APB = 180^\circ - 150^\circ = 30^\circ \]
Step 3: Final Answer:
\( \angle APB = 30^\circ \).