Consider a circle with centre at O and radius 7 cm. Let QR be the 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 triangle OPR to the area of triangle OMP is 1 ∶ 2√2 Select the correct answer using the code given below.?

Question

Consider a circle with centre at O and radius 7 cm. Let QR be the 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 triangle OPR to the area of 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

LohaHridoy

7 views

2025-01-04 04:42

1 Answers

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AhmedNazir · Accepted Answer · 2022-09-04T04:29:57+06:00

Option 1 : 1 and 2 only

⇒ OP² = OQ² – PQ²

⇒ OP = √(7² – 1²) = √48 cm

In triangle OPM

∠OMP = 90°

⇒ OM² = OP² – PM²

⇒ OM = √(48 – 24) = √24 cm

In triangle OMC

∠OMC = 90°

⇒ MC² = OC² – OM²

⇒ MC = √(7² – 24) = √25 = 5cm

Statement I:

In triangle OPM

OM = PM = √24

⇒ It is an isosceles right triangle

⇒ ∠OPM = 45°

∠RPM = ∠RPO + ∠OPM

⇒ ∠RPM = 90 + 45 = 135°

∠QPD = ∠RPM = 135°

⇒ Statement I is true.

Statement II:

CP, m = MC + MP = 5 + √24 cm

PD, n = MD – PM = 5 – √24 cm

⇒ Roots of the equations 5 ± √24

⇒ Sum of roots = 10

⇒ Product of roots = (5 + √24) (5 – √24) = 25 – 24 = 1

⇒ The quadratic equation = x² – 10x + 1 = 0

⇒ Statement II is true

Statement III:

Area of OPR/Area of OMP = (1/2 × PR × OP)/(1/2 × OM × PM)

⇒ (1/2 × 1 × √48)/(1/2 × √24 × √24)

⇒ √

⇒ 1/√12 = 1 ∶ 2√3

⇒ Statement III is wrong

∴ Only Statement 1 and 2 is correct.