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A variable circle passes through the fixed point A(a, b) and touches the x-axis. The other end of the diameter through A traces a locus. What is the equation of this locus?

1. (x - a)² = 4by
2. (y - b)² = 4ax
3. (x - a)² = 2by
4. (x + a)² = 4by

Correct answer: (x - a)² = 4by

Solution

Let P(h, k) be the other diameter end; the centre is ((h+a)/2, (k+b)/2). Touching the x-axis means radius = (k+b)/2, and radius also equals distance from centre to A. Equating gives (h - a)² = 4bk, i.e. (x - a)² = 4by.

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