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Question

Through the vertex O of the parabola y2=4ax,a perpendicular is drawn to any tangent meeting it at P and the parabola T Q find the value of OP .OQ?

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Solution

We have,

Equation of tangent

y2=4ax at point (at2,2at)

Now

yt=x+at2

xyt+at2=0.....(1)

Then, length of perpendicular from O(0,0) is

OP=∣ ∣xyt+at21+t2∣ ∣

OP=∣ ∣0x0t+at21+t2∣ ∣

OP=∣ ∣at21+t2∣ ∣

Now, Equation of perpendicular to (1) is,

tx+y=k

It passes through (0,0)

Now,

0+0=k

k=0

Then,

tx+y=0

y=tx.....(2)

Put in equation of parabola

y2=4ax

t2x2=4ax

x=4at2

From equation

y=tx

y=t4a2t2=4at

Hence, Q=(4at2,4at)

Using distance formula

OQ=(4at20)2+(4at0)2

=4a1t4+1t2

=4a1+t2t4

OP.OQ=at21+t24at21+t2=4a2

Hence, this is the answer.

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