Question

The locus of a point on the variable parabola `y^2=4a x ,` whose distance from the focus is always equal to `k ,` is equal to (`a` is parameter) `4x^2+y^2-4k x=0` `x^2+y^2-4k x=0` `2x^2+4y^2-9k x=0` `4x^2-y^2+4k x=0`

Answer 1

Correct Answer - `4x^(2)+y^(2)-4kx=0`
Let the point be `(at^(2),2at)` and focus S be (a,0).
Now, `SP=at^(2)+a=k` (given) (1)
Let `(alpha,beta)` be the moving point. Then
`alpha=at^(2)andbeta=2at`
`rArr" "(alpha)/(beta)=(t)/(2)anda=(beta^(2))/(4alpha)" "(because "Point" (alpha,beta) "lies on" y^(2)=4ax)`
On substituting these value in equation (1), we get
`(beta^(2))/(4alpha)(1+(4alpha^(2))/(beta^(2)))=k`
`rArr" "beta^(2)+4alpha^(2)=4kalpha`
`rArr" "4x^(2)+y^(2)-4kx=0,`
which is the required locus.