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Question

PN is an ordinate of the parabola ; a straight line is drawn parallel to the axis to bisect NP and meets the curve in Q ; prove that NQ meets the tangent at the vertex in a point T such that AT=23NP, where A is the vertex.

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Solution

The equation of parabola be y2=4ax

let the point P be (at2,2at)

PN is ordinate N(at2,0)

Equation of straight line bisecting NP is

y=at

substituting y in equation of parabola

a2t2=4axx=at24

So the coordinates of Q are (at24,at)

Equation of NQ is

y0=at0at24at2(xat2)y=43t(xat2)

Put x=0

y=43t(0at2)y=4at3

AT=4at3NP=2atATNP=4at32at=23AT=23NP



697732_641432_ans_f61ee3d4b150407ba2180e16f7f4a273.png

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