The centre of the circle S = 0 lie on the line 2x – 2y + 9 = 0 and S = 0 cuts orthogonally the circle x2 + y2 = 4. Show that circle S = 0 passes through two fixed points and find their co-ordinates.
(–4, 4) or
Let circle be
S ≡ x2 + y2 2gx + 2fy + c = 0 …(1)
Since centre of this circle (–g, –f) lie on 2x – 2y + 9 = 0
And the circle S = 0 and x2 + y2 – 4 = 0 cuts orthogonally.
Substituting the values of g and c from (2) and (3) in (1), then
x2 + y2 + (2f + 9)x + 2fy + 4 = 0
or (x2 + y2 + 9x + 4) + 2f (x + y) = 0
hence the circle S = 0 passes through fixed point (
After solving we get (–4, 4) or
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