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: Evaluating Assertion (A):

The assertion posits that a quadrilateral \( OAPB \) formed by tangents \( PA \) and \( PB \) from an external point \( P \) to a circle with center \( O \) is cyclic.
Key geometric properties dictate that tangents from an external point to a circle are equal in length and perpendicular to the radius at the point of contact.
Consequently, \( \angle OAP = \angle OBP = 90^\circ \).
In quadrilateral \( OAPB \), the sum of these opposite angles is:\[\angle OAP + \angle OBP = 90^\circ + 90^\circ = 180^\circ\]This fulfills the condition for a cyclic quadrilateral: a quadrilateral is cyclic if a pair of its opposite angles sum to \(180^\circ\).
Thus, Assertion (A) is substantiated.

Step 2: Evaluating Reason (R):

The reason states: "In a cyclic quadrilateral, opposite angles are equal."
This statement is inaccurate. The correct geometric principle is: in a cyclic quadrilateral, opposite angles are supplementary (their sum is \(180^\circ\)), not necessarily equal.
Therefore, Reason (R) is invalidated.

Step 3: Final Assessment:

- Assertion (A) is verified as true: Quadrilateral \( OAPB \) is indeed cyclic due to the \(180^\circ\) sum of angles at \( A \) and \( B \).
- Reason (R) is determined to be false: Opposite angles in a cyclic quadrilateral are supplementary, not necessarily equal.

Correct Analysis: Assertion (A) is true, while Reason (R) is false.