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Find the equation of the circle that is orthogonal to both circles x² + y² - 4x - 6y + 11 = 0 and x² + y² - 10x - 4y + 21 = 0, and which has the line 2x + 3y = 7 as a diameter.

1. x² + y² - 4x - 2y + 3 = 0
2. x² + y² - 4x - 2y - 3 = 0
3. x² + y² + 4x + 2y + 3 = 0
4. x² + y² - 2x - 4y + 3 = 0

Correct answer: x² + y² - 4x - 2y + 3 = 0

Solution

The two orthogonality equations together with the centre lying on 2x + 3y = 7 give g = -2, f = -1, c = 3, so the circle is x² + y² - 4x - 2y + 3 = 0.

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