A variable chord of the circle $x^2+y^2-2ax = 0$ is a chord of contact of the circle $x^2+y^2 = a^2$. The locus of the midpoint of the variable chord is:
Step-by-Step Solution
Key Concept: Let midpoint be $(h, k)$. Chord of $x^2+y^2-2ax = 0$ with midpoint $(h, k)$: $hx+ky-a(x+h) = h^2 +k^2 -2ah$. This must equal chord of contact $px+qy = a^2$ of $x^2 +y^2 = a^2$ for some $(p, q)$. Match coefficients and eliminate $(p, q)$ to get $h^2 + k^2 = ah$.
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Correct Answer: (1)