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Question

A pair of tangents are drawn from a point on the directrix to a parabola y2=4ax. The angle formed by the tangents will always be 90


A

True

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B

False

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Solution

The correct option is A

True


Let's draw a parabola first

p is the point on the directrix and A and B are the points where tangents from P touches the parabola.

Let P(a,y), both tangents at A point is,

The equation for tangent at A point is,

ty=x+at2

for tangent at A

t1y=a+at21

y=at1(t211) ..............(1)

for tangent at B

t2y=a+at22

y=at2(t221) ......(2)

Eliminating y

t11t1=t21t2

t1t2=1t11t2

=t2t1t1t2

i.e., t1t2=1

We know slope of a tangent at a point given by parameter is (1t)

If 2 tangent are perpendicular then product of slope will be -1.

i.e., 1t1.1t2=1

t1t2=1

Tangent are at right angle to each other


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