If three distinct normals to the parabol...

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

Text Solution

Verified by Experts

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`

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`(-18, -12)`

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`90^(@)`

`60^(@)`

`45^(@)`

`30^(@)`