Question

The abscissae and ordinate of the end points A and B of a focal chord of the parabola y² = 4x are respectively the roots of equations x² – 3x + a = 0 and y² + 6y + b = 0. The equation of the circle with AB as diameter, is –

• A. x² + y² – 3x + 6y + 3 = 0
• B. x² + y² – 3x + 6y – 3 = 0
• C. x² + y² + 3x + 6y – 3 = 0
• D. x² + y² – 3x – 6y – 3 = 0

Answer 1

Answer: (B)

x² + y² – 3x + 6y – 3 = 0

Answer 2

x² – 3x + a = 0 Ž x₁ + x₂ = 3, x₁ x₂ = a

also y² + 6y + b = 0 Ž y₁ + y₂ = –6, y₁y₂ = b

x₁ x₂ = $\frac{1}{t_{1}^{2}}$t₁² = 1 = a

A(x₁, y₁), B(x₂, y₂)

t₁ t₂ = – 1

(t₁² , 2t₁) (t₂², 2t₂) Ž t₂ = $\frac{1}{t_{1}}$

y₁ y₂ = 2t₁ $\left( - \frac{2}{t_{1}} \right)$= – 4 = b

so a = 1, b = –4

Equation of circle

x² + y² – (x₁ + x₂) x – (y₁ + y₂) y + x₁x₂ + y₁y₂ = 0

x² + y² – 3x + 6y – 3 = 0