The normal to the parabola y2= 4ax at the point P(at2 , 2at) meets the x-axis at A. Find the equation of the locus of the midpoint of AP as t varies.
Answer:-
To find the normal;
At A(y,x) but y=0
Since;
Slope of the tangent at
; =
Hence slope of the normal at P is given as;
=
Therefore equation of the normal at P is given by;
Hence;
But y=0
Rewrite as;
Divide by t,we get;
Take the midpoint of AP to be B(h,k)
Here;
=
=
Since ;
We have the equation of the locus as t varies given by;
Using h and k ,we can make the following substitutions;
Rewrite as ;
Write in terms of x and y as;
which is the standard locus at midpoint AP.
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