If two normals drawn from any point to the parabola `y^(2) = 4ax` make angle `alpha` and `beta` with the axis such that tan`alpha` . tan `beta` = 2, then find the locus of this point,

Explanation

Let the point is (h, k). The equation of any normal to the parabola
`y^(2)=4ax` is
`y = mx - 2am - am^(3)`
passes through (h, k)
`!k=mh-2am-am^(3)`
`!am^(3)+m(2a-h)+k=0` …(i)
`m_(1),m_(2),m_(3)` are roots of the equation, then `m_(1),m_(3),m_(3)`
`=-k/a` but `m_(1)m_(2)`
`=2,m_(3)`
`=-k/(2a)`
`m_(3)` is root of (i)
`thereforea(-k/(2a))^(3)-k/(2a-h)+k`
`!k^(2)=4ah`
Thus locus is
`y^(2)=4ax`