AB is diameter of the circle. PA is tangent and ∠AOC = 60°. Assertion(A): x + 30° = 90°. Reason(R): PA is tangent ⇒ ∠BAP = 90° ∴ x + 30° = 90° 1. A is true, R is false. 2. A is false, R is true. 3. Both A and R are true and R is correct reason for A. 4. Both A and R are true and R is incorrect reason for A.

We know that, The angle subtended by an arc of a circle at the center is double the angle subtended by it at any point on the remaining part of the circle. ∴ ∠AOC = 2 x ∠ABC ⇒ 60° = 2 x ∠ABC ⇒ ∠ABC = = 30° The tangent at any point of a circle is perpendicular to the radius through the point of contact. ∴ AP ⊥ OA ⇒ ∠OAP = 90° ⇒ ∠OAP = ∠BAP = 90° In ΔABP, according to angle sum property, ∴ ∠ABP + ∠APB + ∠BAP = 180° ⇒ 30° + x + 90° = 180° ⇒ 30° + x = 180° - 90° ⇒ 30° + x = 90°. ∴ Both A and R are true and R is correct reason for A. Hence, option 3 is the correct option.

Mathematics

AB is diameter of the circle. PA is tangent and ∠AOC = 60°.
Assertion(A): x + 30° = 90°.
Reason(R): PA is tangent
⇒ ∠BAP = 90°
∴ x + 30° = 90°
A is true, R is false.
A is false, R is true.
Both A and R are true and R is correct reason for A.
Both A and R are true and R is incorrect reason for A.

Circles

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Mathematics

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