Find the locus of the point P from which tangents are drawn to the parabola `y^(2) = 4ax` having slopes `m_(1)` and `m_(2)` such that
`theta_(1)-theta_(2)=theta_(0)`(constant)
where` theta_(1)` and `theta_(2)` are the inclinations of the tangents from positive x-axis.

Explanation

Equation of tangent to
`y^(2)=4ax` is
`y=mx+a//m`
Let it passes through P(h, k).
`thereforem^(2)h-mk+a=0`
`theta_(1)-theta_(2)=theta_(0)`
`tan(theta_(1)-theta_(2)=tantheta_(0)`
`!(m_(1)-m_(2))/(1+m_(1)m_(2))=tantheta_(0)`
`!(m_(1)+m_(2))^(2)-4m_(1)m_(2)=tan^(2)theta_(0)(1+m_(1)m_(2))^(2)`
`(k^(2))/(h^(2))-(4a)/h=tan^(2)theta_(0)(1+a/h)^(2)`
`!k^(2)-4ah=(h+a)^(2)tan^(2)theta_(0)`
`therefore` locus of P(h, k) is
`y^(2)-4ax=(x+a)^(2)tan^(2)theta_(0)`