UPSC 2016 CDS I Math Paper

$OP = OQ$ . ....(1)(Lengths of tangents drawn from an external point to a circle are equal.)It is given that:$OR \times SQ = OS \times RP$$\Rightarrow OR \times (OQ - OS) = OS \times (OP - OR)$$\Rightarrow OR \times OQ - OR \times OS = OS \times OP - OS \times OR$$\Rightarrow OR \times OQ = OS \times OP$$\Rightarrow OR = OS$ [using (1)]Since the radius of circle X is equal to the radius of circle Y, so the two circles X and Y are congruent or equal.Hence, statement 1 is correct.Remember: If two tangents are drawn to a circle from an external point, then tangents subtend equal angles at the centre and they are equally inclined to the segment joining the centre to that point$\text{therefore} angle POC = angle QOC = \frac{angle POQ}{2}$....(2)and $angle PCO = angle QCO = \frac{angle PCQ}{2}$...(3)Now $angle POQ + angle PCQ = 180^{\circ}$$\Rightarrow 2 angle POC + 2 angle QCO = 180^{\circ}$ [using (2) and (3)]$\Rightarrow angle POC + angle QCO = 90^{\circ}$Hence, statement 2 is correct.