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  • Two tangents are drawn from a point (2, -1) to a parabola x2 = 4y. Find the angle between the tangentsOption: 1 <img alt="\frac{\pi}{6}" s

Two tangents are drawn from a point (2, -1) to a parabola x2 = 4y. Find the angle between the tangents

Option: 1

\frac{\pi}{6}


Option: 2

\frac{\pi}{2}


Option: 3

\frac{\pi}{3}


Option: 4

\frac{\pi}{4}


Answers (1)

best_answer

As we have learnt,

The combined equation of the pair of tangents drawn from an external point P(x1,y1) to a parabola is  SS1=T2.

 

Now

Combined equation of the tangents from (2,–1) to x2 - 4y = 0 is

(x2 - 4y)(x12 - 4y1) = (xx1 - 2(y + y1))2

(x2 - 4y)(4 + 4) = (2x - 2(y - 1))2

8x2 - 32y = (2x - 2y + 1)2

8x2 - 32y = 4x2 + 4y2 + 1 - 8xy - 4y + 4x

4x2 - 4y2 + 8xy - 4x - 28y - 1 = 0

Here a = 4, b = -4, so a + b = 0

And hence the angle between the lines = 90o

 

Alternate Method

The point (2,-1) lies on the directrix of the given parabola, and we know that the directrix is the director circle of parabola. So the angle between tangents drawn from (2,-1) to this parabola will be 90o

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