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(B) Let the point $P$ on the parabola $y^2 = 4ax$ be $(at^2, 2at)$.The equation of the tangent at $P$ is $ty = x + at^2$ $....(1)$The focus of the par

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$2x^2 + y^2 - 2ax = 0$

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(B) Let the point $P$ on the parabola $y^2 = 4ax$ be $(at^2, 2at)$.The equation of the tangent at $P$ is $ty = x + at^2$ $....(1)$The focus of the parabola is $S(a, 0)$.The line perpendicular to the tangent $(1)$ passing through $S(a, 0)$ has the slope $-t$. Its equation is $y - 0 = -t(x - a)$,which simplifies to $tx + y = at$ $....(2)$The line joining the vertex $O(0, 0)$ to $P(at^2, 2at)$ has the slope $\frac{2at}{at^2} = \frac{2}{t}$. Its equation is $y = \frac{2}{t}x$,or $2x - ty = 0$ $....(3)$To find the locus of $R(h, k)$,we eliminate $t$ from $(2)$ and $(3)$.From $(3)$,$t = \frac{2h}{k}$.Substituting this into $(2)$: $(\frac{2h}{k})h + k = a(\frac{2h}{k})$.Multiplying by $k$: $2h^2 + k^2 = 2ah$.Replacing $(h, k)$ with $(x, y)$,the locus is $2x^2 + y^2 - 2ax = 0$.

The straight line joining any point $P$ on the parabola $y^2 = 4ax$ to the vertex and the perpendicular from the focus to the tangent at $P$ intersect at $R$. Then the equation of the locus of $R$ is:

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