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Question

If the normal to the parabola y2=4ax at the point (at2,2at) cuts the parabola again at (aT2,2aT), then the range of T is

A
T(2,2)
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B
T(,8)(8,)
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C
T(22,22)
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D
T[,22][22,]
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Solution

The correct option is D T[,22][22,]
Given parabola is y2=4ax
Equation of normal at (at2,2at) is
y+tx=2at+at3 (1)

This cuts again at (aT2,2aT), so
2aT+taT2=2at+at32T+tT2=2t+t3 (a0)t(t2T2)+2(tT)=0(tT)(t(t+T)+2)=0t2+Tt+2=0 (tT)
As tR, so
D0T280T[,22][22,]


Alternate solution :
If normal at (at2,2at) again cuts the parabola at (aT2,2aT), then
T=t2tt2+Tt+2=0
As tR, so
D0T280T[,22][22,]

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