Question

A curve passing through the point (1, 1) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the x-axis. Determine the equation of the curve.

Answer 1

The equation of the normal at the point (x, y) is

Y – y = – (dx/dy)(X – x) ...(i)

The distance of perpendicular from the origin to the normal (i) is

gives x = k

which is passing through (1, 1), so k = 1.

Thus, the equation of the curve is x = 1 

Also, dx/dy = 0 = (y² – x²)/2xy

which is a homogeneous differential equation.

Let y = vx

x² + y² = cx

which also passes through (1, 1), so c = 2. 

Hence, the equation of the curve is x² + y² = 2x