Question

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°

विकल्प

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

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 × ∠ABC

⇒ 60° = 2 × ∠ABC

⇒ ∠ABC = `(60°)/2​` = 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.