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Find the equation of the circle passing through the points of intersection of the circle x² + y² - 2x - 4y + 4 = 0 and the line x + 2y = 4, and which touches the line x + 2y = 0.

1. x² + y² - 2x - 4y + 4 = 0
2. x² + y² - 4x - 2y + 4 = 0
3. x² + y² - 2x - 4y + 8 = 0
4. x² + y² - x - 2y + 1 = 0

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

Solution

The family of circles through the intersection is x² + y² - 2x - 4y + 4 + lambda*(x + 2y - 4) = 0; applying the tangency condition to x + 2y = 0 gives lambda = 0, so the required circle is the original one, x² + y² - 2x - 4y + 4 = 0, which indeed touches x + 2y = 0.

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