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

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by ayushk9919

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by ayushk9919

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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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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

=============================================