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: Understanding the Assertion (A):

The assertion states that if \( PA \) and \( PB \) are tangents drawn to a circle with center \( O \) from an external point \( P \), then the quadrilateral \( OAPB \) is a cyclic quadrilateral.
We know from geometry that tangents drawn from an external point to a circle are equal in length and make a right angle with the radius at the point of contact.
So, \( \angle OAP = \angle OBP = 90^\circ \).
Now, consider the sum of these two opposite angles in the quadrilateral \( OAPB \):
\[ \angle OAP + \angle OBP = 90^\circ + 90^\circ = 180^\circ \] This satisfies the property of a cyclic quadrilateral, which states: if a pair of opposite angles of a quadrilateral is supplementary (adds up to \(180^\circ\)), then the quadrilateral is cyclic.
Therefore, Assertion (A) is true.

Step 2: Understanding the Reason (R):

The reason says: "In a cyclic quadrilateral, opposite angles are equal."
This is false because the correct property is: in a cyclic quadrilateral, opposite angles are supplementary — their sum is \(180^\circ\), but they are not necessarily equal.
Hence, Reason (R) is false.

Step 3: Conclusion:

- Assertion (A) is true: The quadrilateral \( OAPB \) is cyclic because the sum of angles at points \( A \) and \( B \) is \(180^\circ\).
- Reason (R) is false: Opposite angles in a cyclic quadrilateral are not always equal; they are supplementary.

Correct Answer: Assertion (A) is true but Reason (R) is false.