Find the locus of the middle points of the chords of the parabola `y^(2) = 4x` which touch the parabola `x^(2) = -8y`.

Explanation

Let the mid-point of a chord of the parabola
`y^(2) = 4x` be (h, k)
`therefore` Equation of the chord is
`T = S_(2)`
`rArr ky - 2(x+h) = k^(2) - 4h`
`rArr ky - 2x = k^(2)-2h`
or `y = (2)/(k)x + (k^(2) - 2h)/(k)" ...(i)`
Line (i) touches the parabola
`therefore (k^(2) -2h)/(k) = +2((2)/(k))^(2)` `{ because "line " y = mx + c " touches the parabola" x^(2) = 4ay " if"c = am^(2)}`
`rArr k^(3) -2hk = 8`
`rArr "Locus of " (h,k) "is" `y_(3) - 2xy - 8 = 0`