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Question

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.

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Solution

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=at2at

Coordinates of point A(a,at2at)

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)=y2y1x2x1=at2at09a=(t21t)2

m1=1t22t

Slope of line PS(m2)=y2y1x2x1=2at0at2a=2tt21

We see that

m1m2=1

=1t22t×2tt21=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.


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