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Question

If the normal at point P(at21,2at1) and Q(at22,2at2) on the parabola y2=4ax meet on the parabola, the t1t2=2.

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Solution

Equation of parabola y2=4ax

And the normal point

P(at12,2at1)

Q(at22,2at2)

Prove that:-

t1t2=2


Proof:-

We know that,

If normal at point t1 to the parabola y2=4ax meets the parabola again at point t3,

Then,

t3=t12t1(Importent)


According to given question,

Normals at point t1 and t2 meet at a point on parabola,

Hence, t3 will be same for them,

t3=t3

t12t1=t22t1

t2t1=2t12t2

(t2t1)=2(t2t1t1t2)

1=2t1t2

t1t2=2


Hence proved.


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