Question

Let a circle $x^2+y^2-2x-3=0$ touches the directrix of a parabola and passes through end points of latus rectum of the same parabola. If latus rectum of the parabola is chord of maximum length with respect to given circle and equation of parabola is $y^2=kx$, then $k=$

Answer 1

For the given parabola directrix will be parallel to $y-$axis
and vertex will be $(0,0)$
The given circle
$x^2+y^2-2x-3=0\Rightarrow (x-1{)}^2+y^2=2^2$
Now the length of the latus rectum to be maximum it should be one of the diameter parallel to $y-$axis and it's mid point will be focus of the parabola
Hence coordinates of the focus will be $(1,0)$
Hence parabola will be
$y^2=4x$
$\therefore k=4$