How many normals can be drawn to parabola `y^(2)=4x` from point (15, 12)? Find their equation. Also, find corresponding feet of normals on the parabola.

Explanation

Equation of the normal to parabola `y^(2)=4x` having slope m is
`y=mx-2m-m^(3)`
If it passes through the point (15, 12), then
`12=15m-2m-m^(3)`
`or" "m^(2)-13m+12=0`
Clearly, one root of the equation is 1.
`:." "(m-1)(m^(2)+m-12)=0`
`rArr" "(m-1)(m-3)(m+4)=0`
`rArr" "m=1,3,-4`
Thus, three normal can be drawn from (15, 12).
The equations of normal and corresponding points on the curve are given by