If x + y = 3 is the equation of the chord AB of the circle x2 + y2 –

1.	If x + y = 3 is the equation of the chord AB of the circle x2 + y2 – 2x + 4y – 8 = 0, find the equation of the circle having bar AB as diameter.
Answer» Required equation of circle passing through intersection S = 0 and L = 0 is S + λL = 0 (x² + y² – 2x + 4y – 8) + λ(x + y – 3) = 0 (x² + y² + x(–2 + λ) + y (4 + λ) – 8 – 3λ = 0 —— (i) x² + y² + 2gx + 2fy + c = 0 —— (ii) Comparing (i) and (ii) we get g = (-2 + λ)/2, f = (4 + λ)/2 Centre lies on x + y = 3 ∴ - ((-2 + λ)/2) - ((4 + λ)/2) = 3 2 – λ – 4 – λ = 6 –2λ = 8 ⇒ λ = – 4 Required equation of circle be (x² + y² – 2x + 4y – 8) – 4(x + y – 3) = 0 x² + y² – 6x + 4 = 0

If x + y = 3 is the equation of the chord AB of the circle x2 + y2 – 2x + 4y – 8 = 0, find the equation of the circle having bar AB as diameter.

Answer»

Required equation of circle passing through intersection S = 0

and L = 0 is S + λL = 0

(x² + y² – 2x + 4y – 8) + λ(x + y – 3) = 0

(x² + y² + x(–2 + λ) + y (4 + λ) – 8 – 3λ = 0 —— (i)

x² + y² + 2gx + 2fy + c = 0 —— (ii)

Comparing (i) and (ii) we get

g = (-2 + λ)/2, f = (4 + λ)/2

Centre lies on x + y = 3

∴ - ((-2 + λ)/2) - ((4 + λ)/2) = 3

2 – λ – 4 – λ = 6

–2λ = 8 ⇒ λ = – 4

Required equation of circle be

(x² + y² – 2x + 4y – 8) – 4(x + y – 3) = 0

x² + y² – 6x + 4 = 0

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If x + y = 3 is the equation of the chord AB of the circle x2 + y2 – 2x + 4y – 8 = 0, find the equation of the circle having bar AB as diameter.

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