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Find the equation of the circle whose centre is the focus of the parabola (x - 1)² = 8y and which touches the parabola at its vertex.

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

Correct answer: x² + y² - 2x - 4y + 1 = 0

Solution

Vertex is (1,0), focus is (1,2) with a = 2; the circle centred at (1,2) touching the vertex has radius 2, giving the stated equation.

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