Consider a circle with centre at and radius 7 cm Let QR be a chord of length 2 cm and let P be the m...........

Quantitative Aptitude (CDS) Coordinate Geometry

Consider a circle with centre at ?? 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 ?? 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 areaof triangle OMP is 1: 2 √2.

Select the correct answer using the code given below.

1 and 2 only

2 and 3 only

1 and 3 only

1, 2 and 3

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Explanation:

CP × PD = QP × PR
⇒ m × n = 1 × 1
⇒ m × n = 1 ... (i)
Now x² – 10 x + 1 = 0
⇒ x =10±√100−4/2
⇒ x = 5±4√3
∴ m×n = (5 + 4√3)(5 − 4√3)
⇒ m×n = 25 − 16×3
⇒ m×n = −23 ... (ii)
From eq. (i) and (ii), we can say that statement 2 is not correct.