O is centre of the circle, PB and PC are tangents and ∠BPC = 50°. Statement (1): ∠BAC = ∠P = 50° Statement (2): ∠BOC + 50° = 180° ⇒ ∠BOC = 130° ∴ ∠BAC = 65° 1. Both the statement are true. 2. Both the statement are false. 3. Statement 1 is true, and statement 2 is false. 4. Statement 1 is false, and statement 2 is true.

The tangent at any point of a circle is perpendicular to the radius through the point of contact. ∴ OB ⊥ BP and OC ⊥ CP ⇒ ∠OBP = 90° and ∠OCP = 90° OCPB is a quadrilateral. ∴ ∠OBP + ∠BPC + ∠OCP + ∠BOC = 360° ⇒ 90° + 50° + 90° + ∠BOC = 360° ⇒ 230° + ∠BOC = 360° ⇒ ∠BOC = 360° - 230° ⇒ ∠BOC = 130° 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. ∴ ∠BOC = 2 x ∠BAC ⇒ 130° = 2 x ∠BAC ⇒ ∠BAC = = 65°. So, Statement 1 is false, and statement 2 is true. Hence, option 4 is the correct option.

Mathematics

O is centre of the circle, PB and PC are tangents and ∠BPC = 50°.
Statement (1): ∠BAC = ∠P = 50°
Statement (2): ∠BOC + 50° = 180°
⇒ ∠BOC = 130°
∴ ∠BAC = 65°
Both the statement are true.
Both the statement are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.

Circles

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Circles

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