Question

In the given figure, PA and PB are tangents to a circle centred at O. If \(\angle AOB = 130^\circ\), then \(\angle APB\) is equal to :

• A. 130^
• B. 50^
• C. 120^
• D. 90^

Hint:

The angle between two tangents drawn from an external point to a circle and the angle subtended by the line segment joining the points of contact at the center are always supplementary.
This means: \(\angle APB + \angle AOB = 180^\circ\).
Therefore, you can directly calculate: \(\angle APB = 180^\circ - 130^\circ = 50^\circ\).
Using this relation saves time during exams!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question requires us to find the angle between two tangents drawn from an external point to a circle, given the angle subtended by the radii at the center.
The tangents are \(PA\) and \(PB\), contacting the circle at points \(A\) and \(B\) respectively.
We are given that the angle subtended at the center \(O\) is \(\angle AOB = 130^\circ\).
We need to determine the value of \(\angle APB\).

Step 2: Key Formula or Approach:
1. A key theorem in circle geometry states that a tangent at any point on a circle is perpendicular to the radius through the point of contact.
Therefore, \(\angle OAP = 90^\circ\) and \(\angle OBP = 90^\circ\).
2. The figure \(OAPB\) forms a quadrilateral.
3. The sum of all interior angles of any quadrilateral is \(360^\circ\).
This geometric approach will help us directly calculate the unknown angle.

Step 3: Detailed Explanation:
1. Let \(O\) be the center of the circle, and let \(PA\) and \(PB\) be the tangents contacting the circle at points \(A\) and \(B\).
2. Since radius is perpendicular to the tangent at the point of contact:
\[ \angle OAP = 90^\circ \]
\[ \angle OBP = 90^\circ \]
3. In quadrilateral \(OAPB\), the sum of the angles is:
\[ \angle OAP + \angle APB + \angle OBP + \angle AOB = 360^\circ \]
4. Substituting the known values into the equation:
\[ 90^\circ + \angle APB + 90^\circ + 130^\circ = 360^\circ \]
\[ 310^\circ + \angle APB = 360^\circ \]
5. Solving for \(\angle APB\):
\[ \angle APB = 360^\circ - 310^\circ = 50^\circ \]
6. Thus, the angle between the tangents is \(50^\circ\).

Step 4: Final Answer:
Hence, the correct option is (B).