Question

Find the equation of line which is normal to the parabola `x^(2)=4y` and touches the parabola `y^(2)=12x`.

Answer 1

Normal to parabola `x^(2)=4y` having slope m is
`y=mx+2+(1)/(m^(2))` (1)
It is tangent to `y^(2)=12x`.
Now, tangent to above parabola having slope m is
`y=mx+(3)/(m)` (2)
Comparing (1) and (2), we get
`(1)/(m^(2))+2=(3)/(m)`
`rArr" "2m^(2)-3m+1=0`
`rArr" "(2m-1)(m-1)=0`
`rArr" "m=(1)/(2)orm=1`
Therefore, equations of lines are 2y=x+12 or y=x+3.