Question

Assertion (A): If the PA and PB are tangents drawn to a circle with center O from an external point P, then the quadrilateral OAPB is a cyclic quadrilateral.
Reason (R): In a cyclic quadrilateral, opposite angles are equal.

• A. Both, Assertion (A) and Reason (R) are true. Reason (R) explains Assertion (A) completely.
• B. Both, Assertion (A) and Reason (R) are true. Reason (R) does not explain Assertion (A).
• C. Assertion (A) is true but Reason (R) is false.
• D. Assertion (A) is false but Reason (R) is true.

Answer 1

The Correct Option is C

Step 1: Verifying the assertion (A):
The assertion states that tangents drawn from an external point to a circle are equal in length, and the angle between the tangent and the radius at the point of contact is \( 90^\circ \).
- This is a well-known property of tangents. If \( P \) is an external point and \( PA \) and \( PB \) are the tangents drawn to the circle from \( P \), then \( PA = PB \). Also, the radius at the point of contact is perpendicular to the tangent. Hence, the assertion is true.
Additionally, the assertion mentions that the quadrilateral formed by the tangents and the radii from point \( P \) is cyclic because the sum of opposite angles equals \( 180^\circ \).
- A cyclic quadrilateral is one in which the sum of the opposite angles is \( 180^\circ \). This is a property of cyclic quadrilaterals, and since the tangents and radii form a cyclic quadrilateral, the assertion is valid.

Step 2: Verifying the reason (R):
The reason states that in a cyclic quadrilateral, opposite angles are supplementary, meaning the sum of opposite angles is always \( 180^\circ \).
- This is a fundamental property of cyclic quadrilaterals. By definition, if a quadrilateral is cyclic, the sum of the opposite angles is always \( 180^\circ \). Hence, the reason is true.

Step 3: Conclusion:
Both the assertion (A) and the reason (R) are true. Furthermore, the reason (R) correctly explains the assertion (A), as the sum of opposite angles being \( 180^\circ \) is what makes the quadrilateral cyclic.
Therefore, the correct answer is:
(a) Both assertion and reason are true, and the reason explains the assertion.