Can anyone prove the property that that sum of slopes of normals at conormal points of parabola is 0?
Thnx to anyone who answers!:)
Thnx to anyone who answers!:)
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Consider the parametric equation of a normal to the parabola y^2=4ax
y+xt=2at+at^3 (derived by taking everypoint of the parabola as [at^2,2at] )
Now as t=-m where m is the slope at the point:
The equation becomes: am3+(2a-x)m+y=0
Notice its a third degree polynomial and the coefficient of the 2nd degree term is zero.
This implies that the sum of roots of the equation is zero.
Hence sum of slopes=m1+m2+m3=0
y+xt=2at+at^3 (derived by taking everypoint of the parabola as [at^2,2at] )
Now as t=-m where m is the slope at the point:
The equation becomes: am3+(2a-x)m+y=0
Notice its a third degree polynomial and the coefficient of the 2nd degree term is zero.
This implies that the sum of roots of the equation is zero.
Hence sum of slopes=m1+m2+m3=0