The locus of a point on the variable parabola $y^2 = 4ax$, whose distance from focus is constant $k$, is equal to: ($a$ is parameter)
Step-by-Step Solution
Key Concept: Convert the parametric constraint on a parabola into a Cartesian equation by eliminating the parameter.
Given point $P(at^2, 2at)$ lies on parabola $y^2 = 4ax$, and $SP = at^2 + a = k$, we treat $(a,\beta)$ as the moving point where $a = at^2$ and $\beta = 2at$. Substituting into the constraint gives $\beta^2/4a (1 + 4a^2/\beta^2) = k$, which simplifies to $4x^2 + y^2 - 4kx = 0$ as the required locus.
Correct Answer: 1