Question

IF three distinct normals to the parabola `y^(2)-2y=4x-9` meet at point (h,k), then prove that `hgt4`.

Answer 1

Given parabola is `(y-1)^(2)=4(x-2)`.
Equation of normal to parabola having slope m is
`y-1=m(x-2)-2m-m^(3)`
It passes through (h,k)
`:." "k-1=m(h-2)-2m-m^(3)`
`or" "m^(3)+(4-h)m+k-1=0`
This equation has three distinct real roots.
`:." "3m^(2)+(4-h)=0` has two distinct real roots.
`:." "4-hlt0orhgt4`