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Question

If the normals at A(t1) and B(t2) meet again at C(t3) on the parabola y2=4ax, then the locus of the mid point of AB is

A
y2=2axa2
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B
y2=2ax+4a2
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C
y2=2ax4a2
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D
y2=2ax+a2
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Solution

The correct option is B y2=2ax+4a2
If normal at A(t1) meets again at C(t3) on the parabola y2=4ax, then
t3=t12t1 (i)

If normal at B(t2) meets again at C(t3) on the parabola, then
t3=t22t2 (ii)

From equations (i) and (ii),
t12t1=t22t2t1t2=2 (iii)

Let (h,k) be the mid point of AB, then
h=a(t21+t22)2
2ha=t21+t22 (iv)
and k=a(t1+t2)
k2a2=t21+t22+2t1t2

From equations (iii) and (iv), we get
k2a2=2ha+4
Hence, required locus is y2=2ax+4a2

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