Question

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

• A. Latus rectum is half the latus rectum of the original parabola
• B. Vertex is $(\frac{a}{2},0)$
• C. Directrix is $y-$axis
• D. Focus has the co-ordinates $(a,0)$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A Latus rectum is half the latus rectum of the original parabola
B Vertex is $(\frac{a}{2},0)$
C Directrix is $y-$axis
D Focus has the co-ordinates $(a,0)$

Any point on the parabola is
$P(a{t}^2,2at)$
Therefore, the midpoint of $S(a,0)$ and $P(a{t}^2,2at)$ is,
$R(\frac{a+a{t}^2}{2},at)\equiv (h,k)$
$\Rightarrow h=\frac{a+a{t}^2}{2},k=at$
Eliminating $t$,
$2x=a(1+\frac{y^2}{a^2})=a+\frac{y^2}{a}$
$\Rightarrow 2ax=a^2+y^2$
$\Rightarrow y^2=2a(x-\frac{a}{2})$
It is a parabola with vertex at $(\frac{a}{2},0)$ and latus rectum 2a.
The directrix is
$x-\frac{a}{2}=-\frac{a}{2}$
$\Rightarrow x=0$
The focus is
$x-\frac{a}{2}=\frac{a}{2}$
$\Rightarrow x=a$
Therefore the focus is $(a,0)$