The locus of the midpoint of the focal radii | vedclass

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(D) Let the variable point on the parabola $y^2 = 4ax$ be $P(at^2, 2at)$.The focus of the parabola is $S(a, 0)$.The focal radius is the segment $SP$.

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All of the above

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(D) Let the variable point on the parabola $y^2 = 4ax$ be $P(at^2, 2at)$.The focus of the parabola is $S(a, 0)$.The focal radius is the segment $SP$. The midpoint $M(x, y)$ of $SP$ is given by:$x = \frac{at^2 + a}{2}, y = \frac{2at + 0}{2} = at$.From $y = at$,we get $t = \frac{y}{a}$.Substituting $t$ into the equation for $x$:$x = \frac{a(\frac{y}{a})^2 + a}{2} = \frac{\frac{y^2}{a} + a}{2} = \frac{y^2 + a^2}{2a}$.$2ax = y^2 + a^2 \implies y^2 = 2ax - a^2 = 2a(x - \frac{a}{2})$.This is a parabola with vertex at $(\frac{a}{2}, 0)$.The latus rectum of this new parabola is $4A = 2a$,which is half the latus rectum of the original parabola $(4a)$.The directrix is $x - \frac{a}{2} = -\frac{a}{2} \implies x = 0$,which is the $y$-axis.Thus,all the given statements are correct.

The locus of the midpoint of the focal radii of a variable point moving on the parabola $y^2 = 4ax$ is a parabola whose:

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