UPSC CDS Math Circle Questions

OP = OQ . ....(1)
(Lengths of tangents drawn from an external point to a circle are equal.)
It is given that:
OR × SQ = OS × RP
⇒ OR × (OQ – OS) = OS × (OP – OR)
⇒ OR × OQ – OR × OS = OS × OP – OS × OR
⇒ OR × OQ = OS × OP
⇒ 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
∴∠POC=∠QOC=

∠POQ

2

....(2)
and ∠PCO=∠QCO=

∠PCQ

2

...(3)
Now ∠POQ+∠PCQ=180°
⇒2∠POC+2∠QCO=180° [using (2) and (3)]
⇒∠POC+∠QCO=90°
Hence, statement 2 is correct.