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A circle S = 0 has its centre on the line 2x - 2y + 9 = 0 and cuts the circle x² + y² = 4 orthogonally. Show that S = 0 passes through two fixed points, and find their coordinates.
- (-1/2, 1/2) and (-4, 4)
- (1/2, -1/2) and (4, -4)
- (-1/2, 1/2) and (4, 4)
- (2, -2) and (-2, 2)
Correct answer: (-1/2, 1/2) and (-4, 4)
Solution
The family reduces to (x² + y² - 9y + 4) + 2g(x + y) = 0, so all such circles pass through the intersection of x + y = 0 with x² + y² - 9y + 4 = 0, namely (-1/2, 1/2) and (-4, 4).
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