The portion of a tangent to a parabola cut off between the directrix and point of contact on the curve subtends _____ degree angle at the focus.
Let's consider a standard parabola y2=4ax and a point P(at2,2at)
Equation of tangent
T=0
y.(2at)=2a(x+at2)
ty=x+at2 ...............(1)
Equation of directrix
x=−a ...............(2)
Intersection of tangent & directrix
ty=−a+at2
y=at2−at
Coordinates of point A(−a,at2−at)
Now, we have coordinates of A,P & S.
We can find the slope of line AS & PS, and then find the angle ASP.
Slope of line AS(m1)=y2−y1x2−x1=at2−at−0−9−a=(t2−1t)−2
m1=1−t22t
Slope of line PS(m2)=y2−y1x2−x1=2at−0at2−a=2tt2−1
We see that
m1m2=−1
=1−t22t×2tt2−1=−1
So, we can say that ∠ASP=90∘ OR the portion of atangent
to a parabola cut off between the directrix and point of
contact on the curve subtends a right angle at focus.