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Question

If the normal at two points P and Q of a parabola y2=4ax intersect at a third point R on the curve, then the product of ordinates of P and Q is

A
4 a2
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B
2 a2
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C
4 a2
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D
8 a2
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Solution

The correct option is D 8 a2
Let (at2,2at1),(at22,2at2),(at23,2at3) be coordinates of the point P, Q, R respecting which lie on the curve y2=4ax.

Now, equation of the normal to the curve at P is,

y2at1=2at12a(xat21) ……….(i)

R is a point which lies at the normal at P.

Substituting y=2at3, x=at23 in equation (1)

2at32at1=t1(at23at21)

2a(t3t1)=at1(t23t21)

2(t3t1)=t1(t3t1)(t3+t1)

2=t1(t3+t1)

2=t1t3t21 ……….(ii)

Now, the equation of the normal to the curve at Q is

y2at2=2at22a(xat22) ………..(iii)

R is a point which lies at the normal at Q.

Substituting y=2at3,x=at23 in equation (iii)

2at32at2=t2(at23at22)

2a(t3t2)=at2(t3t2)(t3+t2)

2=t2(t3+t2)

2=t2t3t22 ……..(iv)

Multiplying equation (ii) with t2,

2t2=t1t2t3t21t2 ………..(v)

Multiplying equation (iv) with t1,

2t1=t1t2t3t22t1 …….(vi)

Subtracting (v) and (vi), we get

2t22t1=t21t2+t22t1

2(t2t1)=t1t2(t1t2)

2(t2t1)=t1t2(t2t1)

t1t2=2

Now, the product of ordinates of P and Q=2at12at2

=4a2t1t2

=4a2(2)

=8a2.

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