The tangent and normal at `P(t)`, for all real positive `t`, to the parabola `y^2= 4ax` meet the axis of the parabola in `T` and `G` respectively, then the angle at which the tangent at `P` to the parabola is inclined to the tangent at `P` to the circle passing through the points `P, T `and `G` is

Explanation

Tangent and normal at (t) on the parabola meet the x-axis at T and N, respectively.
We know that SP=SN=ST
So, for circumcircle of triangle of NPT, TN is diameter and S is the centre.
SP is normal to the circle. If `theta` is the angle between tangents at P to the parabola and circle then `(90^(@)-theta)` is the angle between PT and SP.
Slope of tangent `PT=(1)/(t)`
Slope of SP `=(2at)/(at^(2)-a)=(2t)/(t^(2)-a)=(2t)/(t^(2)-1)`
`:." "tan(90^(@)-theta)=|((1)/(t)-(2t)/(t^(2)-1))/(1+(1)/(t)*(2t)/(t^(2)-1))|=(1)/(t)`
`rArr" "cottheta=(1)/(t)`
`rArr" "tantheta=t" "rArr" "theta=tan^(-1)t`