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Question

Tangents are drawn from any point on the line x+4a=0 to the parabola y2=4ax. Then the angle subtended by the chord of contact at the vertex will be .

A
π2
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B
π3
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C
π4
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D
π6
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Solution

The correct option is A π2
Equation of a tangent of the parabola y2=4ax having slope m is y=mx+am.
Let there be a point P(4a, y1) on the line x+4a=0.
If the tangent y=mx+am passes through (4a, y1), then y1=4am+am 4am2+my1a=0
Let the roots of these equation be m1, m2.
Differentiating the given equation of parabola, we get 2yy'=4a y=2ay'
y-coordinates of points say A, B from where these tangents are drawn are 2am1, 2am2
We have to find the angle between OA and OB where O is the origin.
y2=4ax yx=4ay
Slope of a line passing through origin and a point on the parabola(x1, y1) is 4ay1.
Angle between the lines of slopes m1, m2 is tan1((m1m21+m1m2).
Here, slopes are 4a2am1=2m1, 4a2am2=2m2.
So, the angle subtended is tan1((2m12m21+4m1m2).
m1m2=14 (Product of the roots)
Denominator becomes zero.
So, the angle subtended is 900.

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