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Question

For what point of the parabola y2=4ax is (1) the normal equal to twice the sub-tangent, (2) the normal equal to the difference between the sub-tangent and the sub-normal ?

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Solution

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

Length of sub tangent PQ=2at2

Length of subnormal MN=2a

Length of normal PN=2a1+t2

(1) Length of sub tangent = length of normal

2at2=2a1+t2t2=1+t2t4=1+t2t4t21=0

using quadratic formula

t2=1±14(1)(1)2t2=1±52t2=1+52t=1+52

Point of contact is (at2,2at)

a1+52,2a1+52(a+a52,a2+25)

(2) normal = difference between sub tangent and su normal

y2=4ax(at2,2at)

Given : 2a1+t2=2at22a

1+t2=t211+t2=t4+12t2t43t2=0t2(t23)=0t23=0t2=3t=3

Point of contact is (at2,2at)

(3a,23a)


697808_641132_ans_45eb4c1518384143afe3d70325f808be.jpg

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