Consider a circle with centre at C Let OP OQ denote respectively the tangents to the circle drawn fr...........

Quantitative Aptitude (CDS) Coordinate Geometry

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 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.

1 only

2 only

Both 1 and 2

Neither 1 nor 2

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Explanation:

Given a circle with centre C.
OP and OQ are tangents to the circle from Q + O,point O outside the circle.
Given OR × SQ = OS × RP
⇒OR/RP=OS/SQ
RS || PQ (By Basic proportionality theorem)
1. Also CP = CQ = radius of the circle. A perpendicular drawn from P to Q. Draw circle X and Y with centre O and radius OR and DS respectively.
Since RS || PQ
Here O is the center of circle X and Y both Radius OR and OS lies in the same circle.
⇒ OR = OS ⇒ Area of Circle X = Area of circle Y
⇒ X = Y
Statement (1) is true.
2. Also, we know that if two tangents are drawn to the circle, then ∠POC = ∠QOC and ∠PCO = ∠QCO
Also, we know that CP = CQ = radius
So ∠PtC and ΔQtC are similar by AA similarly.
i.e., ∠P = ∠Q = 45° ...(1)
and ∠t = 90°
Also ∠PCO = ∠QOC (Alternate angles)
∠POC = ∠QCO (Alternate angles)
From (1) if ∠P = ∠Q = 45°
∠QCO = ∠PCO = 45°
∠POC + ∠QCO = 45° + 45° = 90°
Statement (2) is true.