Q: The centre of the circle S = 0 lies on the line 2x – 2y + 9 = 0 and S = 0 cuts orthogonally the circle x^{2} + y^{2} = 4. Show that S = 0 passes through two fixed points and find their coordinates.

Solution: Let the circle S = 0 be represented by

x^{2} + y^{2} + 2gx + 2fy + c = 0

Its centre is (-g, -f)

⇒ 2(-g) -2(-f) + 9 = 0

⇒ 2g – 2f = 9 ….(i)

The circle S = 0 cuts the circle x2 + y2 – 4 = 0 orthogonally. Hence

2g × 0 + 2×f×0 = c – 4 Þ c = 4 …(2)

⇒ S ≡ x^{2} + y^{2} + (9 + 2f)x + 2fy + 4 = 0

⇒ x^{2} + y^{2} + 9x + 4 + 2f(x + y) = 0

This passes through the points of intersection of the line

x + y = 0 and the circle x^{2} + y^{2} + 9x + 4 = 0

⇒ x^{2} + (-x)^{2} + 9x + 4 = 0

⇒ 2x^{2} + 9x + 4 = 0

⇒ x = -1/2, -4

⇒ The points are (–1/2, 1/2) and (–4, 4).

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