Consider a circle with centre at O and radius 7 cm. Let QR be a chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √(24) cm, then which of the following statements are correct?1. ∠QPD = 135° 2. If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x² - 10x + 1 = 0. 3. The ratio of the area of triangle OPR to the area of triangle OMP is 1: 2√(2). Select the correct answer using the code given below.
- A1 and 2 only
- B2 and 3 only
- C1 and 3 only
- D1, 2 and 3
Explanation
OP = √(49−1) = √48 and MP = √24 with OM ⊥ CD gives OM = √(49−25) = √24, so ΔOMP is right-isosceles: ∠MPO = 45°, and angle-chasing gives ∠QPD = 135° (statement 1 true). With CP+PD = CD = 10 and CP·PD = PQ·PR = 1, m and n are roots of x²−10x+1 = 0 (statement 2 true). But area(ΔOPR) = 2√3 and area(ΔOMP) = 12, a ratio of 1:2√3, not 1:2√2 (statement 3 false). Hence 1 and 2 only, option A.