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Question

If P and Q are the points of contact of tangents drawn from the point T to y2=4ax and PQ be a normal of the parabola at P, then the locus of the point which bisects TP is

A
x+a=0
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B
x+2a=0
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C
x=0
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D
x=1
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Solution

The correct option is A x+a=0
Given parabola is y2=4ax
Let P(at21,2at1) and Q(at22,2at2), then
T=(at1t2,a(t1+t2))


Let R(h,k) be the mid point of TP

(h,k)=(at1t2+at212,a(t1+t2)+2at12)
If normal at P(t1) intersect the parabola again at Q(t2), then
t2=t12t2t1(t1+t2)=2t21+t1t2=2

Now,
h=2a2=a
Hence, the locus of the midpoint is
x+a=0
This is equation of the directrix of the parabola.

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