Find the arithmetic mean of the ordinates of the feet of the normals from the point (3,5) to the parabola y^2 = 8x

Question

Sunil Raikwar · Answer

Sum of the ordinates of conormal points is zero. we can prove by following approach:
Equation of normal at any point(2t^2,4t) to the parabola y^2=8x is
y= -ty+4t+2t^3
it passes through (3,5)
5= -3t+4t+2t^3
2t^3-t-5+0
since it is a cubic so t can have 3 possible values so three normals can be draw to
the parabola. let roots of this eq. be t1, t2, t3
t1+t2+t3=0
4t1+4t2+4t3=0
y1+y2+y3+0

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Find the arithmetic mean of the ordinates of the feet of the normals from the point (3,5) to the parabola y^2 = 8x

Find the arithmetic mean of the ordinates of the feet of the normals from the point (3,5) to the parabola y^2 = 8x

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