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Consider a circle with centre at C. Let OP, OQ denote respectively the tangents to the circle drawn from a point O outside the circle. Let R be a point on OP and S be a point on OQ such that OR × SQ = OS × RP. Which of the following statement(s) is/are correct?
(1) If X is the circle with centre at O and radius OR, and Y is the circle with centre at O and radius OS, then X = Y
(2) ∠POC + ∠QCO = 90°
Select the correct answer using the code given below.
  • a)
    1 only
  • b)
    2 only
  • c)
    Both 1 and 2
  • d)
    Neither 1 nor 2
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Consider a circle with centre at C. Let OP, OQ denote respectively the...

OR × SQ = OS × RP
⇒ OR/RP = OS/SQ
Adding 1 on both sides
⇒ OR/RP + 1 = OS/SQ + 1
⇒ (OR + RP) /RP = (OS + SQ) /SQ
⇒ OP/RP = OQ/SQ
OP = OQ [∵ Two tangents drawn from same external point are equal in length]
⇒ RP = SQ
⇒ OP – RP = OS – SQ
⇒ OR = OS
Statement I:
Both X and Y are circles with same centre and radius
⇒ X = Y
⇒ Statement I is true
In quadrilateral OPCQ
⇒ ∠O + ∠P + ∠Q + ∠C = 360° [∵ Angle sum property of quadrilateral]
⇒ 2∠POC + 90° + 90° + 2∠QCO = 360 [∵ Tangent is perpendicular to the radius and line joining the centre to the external points bisects the angle between tangents and radius for a pair of tangent drawn from that external point]
⇒ 2∠POC + 2∠QCO = 180°
⇒ ∠POC + ∠QCO = 90°
⇒ Statement II is true
∴ Both I and II are true.
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Consider a circle with centre at C. Let OP, OQ denote respectively the tangents to the circle drawn from a point O outside the circle. Let R be a point on OP and S be a point on OQ such that OR × SQ = OS × RP. Which of the following statement(s) is/are correct?(1) If X is the circle with centre at O and radius OR, and Y is the circle with centre at O and radius OS, then X = Y(2) ∠POC + ∠QCO = 90°Select the correct answer using the code given below.a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer?
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