Q. 333.7( 3 Votes )

In Fig. 10.95, PQ and PR are tangents drawn from P to a circle with centre O. If OPQ = 35°, then


A. a = 30°, b = 60°

B. a = 35°, b = 55°

C. a = 40°, b = 50°

D. a = 45°, b = 45°

Answer :

Given:


OPQ = 35°


Property 1: If two tangents are drawn to a circle from one external point, then their tangent segments (lines joining the external point and the points of tangency on circle) are equal.


Property 2: The tangent at a point on a circle is at right angles to the radius obtained by joining center and the point of tangency.


Property 3: The sum of all angles of a triangle = 180°.


By property 1,


QP = QR (tangent from Q)


By property 2, ∆OPQ is right-angled at OQP (i.e., OQP = 90°) and ORP is right-angled at ORP (i.e., ORP = 90°).


OQ QP


OR RP


Now,


OQP = ORP = 90° [Property 1]


QP = QR [Property 2]


OP = OP [Given]


OPQ OPR By SAS


Hence, OPQ = OPR = 35° By CPCTC


i.e. a = 35°


By property 3,


OQP + OPQ + QOP = 180°


90° + 35° + QOP = 180°


125° + QOP = 180°


QOP = 180° - 125°


QOP = 55°


i.e. b = 55°


Hence, a = 35° and b = 55°

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