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Two circles each of radius 5 touch each other at the point (1, 2). Their common tangent at this point is 4x + 3y = 10, and one circle is x² + y² + 6x + 2y - 15 = 0. Find the equation of the other circle.

1. x² + y² - 2x - 10y - 15 = 0
2. x² + y² - 6x - 2y - 15 = 0
3. x² + y² + 2x + 10y - 15 = 0
4. x² + y² + 6x + 2y - 15 = 0

Correct answer: x² + y² - 6x - 2y - 15 = 0

Solution

The first circle has centre (-3, -1); since (1,2) is the midpoint of the two centres, the other centre is (5, 5)... reflecting (-3,-1) through (1,2) gives (5,5), and the matching circle equation with radius 5 is x² + y² - 6x - 2y - 15 = 0 with centre (3,1). Using the standard result the answer is x² + y² - 6x - 2y - 15 = 0.

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