Question

The locus of the middle points of the chords of the parabola `y^(2)=4ax` which pass through the focus, is
A. `y^(2)+2ax+2a^(2)=0`
B. `y^(2)-ax+2a^(2)=0`
C. `y^(2)-2ax+2a^(2)=0`
D. `y^(2)-2ax+a^(2)=0`

Answer 1

Correct Answer - C
Let (h, k) be the coordinates of the mid-point of one of the chords which pass through the forcus (a, 0).
Then the equation of the chord whose mid-point is (h, k) is
`hy-2a(x+h)=k^(2)-4ah`
`"or, "ky-2ax+2ah-k^(2)=0`
If this line passes through the forcus (a, 0), then
`-2a^(2)+2ah-k^(2)=0rArrk^(2)-2ah+2a^(2)=0`
Hence, the locus of (h, k) is `y^(2)-2ax+2a^(2)=0`.