The portion of a tangent to a parabola cut off between the directrix and the curve subtends right angle at the focus.
True
Here we are required to find the angle betwwen PS and DS.
D is the point where the tangent at P meets the directrix.
We know that the equation of the tangent,
ty=x+at2
The tangent passes through a point D(-a,y') situated on the directrix
ty′=−a+at2
y′=a.(t2−1)t
D≡(−a,a(t2−1)t)
Slope of SD=a(t2−1)t(−2a)=−t2−12t=m1
Slope of PS=2ata(t2−1)=2tt2−1=m2
∴ m1×m2=−1
∴ The time segment PD extends right angle at the focus S.