Question

Find alll curves in the xy plane for which the normal at each point (x,y) intersects the x axis at (1+x,0).

Answer 1

The normal to the curve y = f(x) at P (x1, y1) intersects the x axis at (1+x1 , 0).
Let the normal be :  y = m x + c

then,  y1 = m x1 + c
and    0 = m (1+x1) + c    =>   c = - m - m x1
Then  y1 = -m, 

m = slope of normal at P(x, y) = - y
So, Slope of tangent at P(x,y) = 1/y  = d y / d x

So,  d y/ d x = 1/ y  or,    y dy = dx    then 
integrating both sides,          y² /2 = x + K1    or y² = 2 x + K
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