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: Assertion (A) Verification:
The assertion is that tangents from an external point to a circle are of equal length and form a \( 90^\circ \) angle with the radius at the point of contact.
- This is a standard geometric property. If \( P \) is an external point and \( PA \) and \( PB \) are tangents to a circle from \( P \), then \( PA = PB \). The radius is also perpendicular to the tangent at the point of contact. Thus, the assertion is valid.

The assertion also states that the quadrilateral formed by these tangents and radii from \( P \) is cyclic due to opposite angles summing to \( 180^\circ \).
- A cyclic quadrilateral's opposite angles sum to \( 180^\circ \). This property applies here, confirming the assertion's validity.

Step 2: Reason (R) Verification:
The reason states that in a cyclic quadrilateral, opposite angles are supplementary (sum to \( 180^\circ \)).
- This is the definition of a cyclic quadrilateral. Therefore, the reason is true.

Step 3: Conclusion:
Both assertion (A) and reason (R) are true. Reason (R) correctly justifies assertion (A) because the property of opposite angles summing to \( 180^\circ \) is what defines a cyclic quadrilateral, which is formed in this scenario.
The correct answer is:
(a) Both assertion and reason are true, and the reason explains the assertion.