The locus of the midpoint of the focal distance of a variable point moving on the parabola, $y^2 = 4ax$ is a parabola whose
Step-by-Step Solution
Key Concept: For a variable point P(at², 2at) on parabola y² = 4ax with focus S(a, 0), the midpoint R of SP traces a new parabola y² = 2a(x - a/2). This locus parabola has half the latus rectum (2a vs 4a) and its directrix coincides with the original parabola's directrix (y-axis or x = 0).
Any point on the parabola is $P(at^2, 2at)$. The midpoint of $S(a, 0)$ and $P$ is $R(\frac{a + at^2}{2}, at)$. Setting $h = \frac{a + at^2}{2}$ and $k = at$, we eliminate $t$ to get $y^2 = 2a(x - \frac{a}{2})$. This is a parabola with vertex at $(\frac{a}{2}, 0)$, latus rectum $2a$, directrix $x = 0$, and focus at $(a, 0)$.
Correct Answer: 1,2,3,4