The locus of the midpoint of the focal distance of a variable point moving on theparabola `y^2=4a x` is a parabola whose (a)latus rectum is half the latus rectum of the original parabola (b)vertex is `(a/2,0)` (c)directrix is y-axis. (d)focus has coordinates (a, 0)

Options

A. latus rectum is half the latus rectum of the original parabola
B. vertex is (a/2,0)
C. directrix is y-axis
D. focus has coordinates (a,0)

Answer

A, B, C, D

Explanation

1,2,3,4
Any point on the parabola is `P(at^(2),2at)`.
Therefore, the midpoint of S(a,0) and `P(at^(2),2at)` is
`R((a+at^(2))/(2),at)-=(h,k)`
`:.h=(a+at^(2))/(2),k=at`
Eliminate t, i.e.,
`2x=a(1+(y^(2))/(a^(2)))=a+(y^(2))/(a)`
`i.e.," " 2ax=a^(2)+y^(2)`
`i.e," "y^(2)=2a(x-(a)/(2))`
It is a parabola with vertex at (a/2,0) and latus rectum 2a.
The directrix is
`x-(a)/(2)=-(a)/(2)`
`i.e," "x=0`
The focus is
`x-(a)/(2)=(a)/(2)`
i.e, x=a
i.e., (a,0)