Find the equation of a circle which bisects the circumferences of the

1.	Find the equation of a circle which bisects the circumferences of the circles x2 + y2 = 1, x2 + y2 + 2x = 3 and x2 + y2 + 2y = 3.
Answer» Let the given circles be C₁, C₂ and C₃, respectively. Let C be the required circle and its equation be S ≡ x² + y² + 2gx + 2fy + c = 0. Let S' ≡ x² + y² +1 = 0. Since C bisects the circumference of C₁, the line S − S' ≡ 2gx + 2fy + c +1 = 0 passes through the centre of C₁ = (0, 0). Therefore, c = 1 Now, C bisects the circumference of C₂ the line S − S'' ≡ 2(g − 1)x + 2fy + c +3 = 0 passes through the centre (−1, 0) of C₂. Therefore 2(g - 1)(-1) - 1 + 3 = 0 -2g + 2 - 1 + 3 = 0 g = 2 Similarly, since the circle C bisects the circumference of circle C₃, we have f = 2. Therefore, equation of the required circle C is S ≡ x² + y² + 4x + 4y - 1 = 0

Find the equation of a circle which bisects the circumferences of the circles x2 + y2 = 1, x2 + y2 + 2x = 3 and x2 + y2 + 2y = 3.

Answer»

Let the given circles be C₁, C₂ and C₃, respectively. Let C be the required circle and its equation be S ≡ x² + y² + 2gx + 2fy + c = 0. Let S' ≡ x² + y² +1 = 0. Since C bisects the circumference of C₁, the line S − S' ≡ 2gx + 2fy + c +1 = 0 passes through the centre of C₁ = (0, 0). Therefore,

c = 1

Now, C bisects the circumference of C₂

the line S − S'' ≡ 2(g − 1)x + 2fy + c +3 = 0 passes through the centre (−1, 0) of C₂. Therefore

2(g - 1)(-1) - 1 + 3 = 0

-2g + 2 - 1 + 3 = 0

g = 2

Similarly, since the circle C bisects the circumference of circle C₃, we have f = 2. Therefore, equation of the required circle C is

S ≡ x² + y² + 4x + 4y - 1 = 0

Discussion

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