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 any point P(x, y) is

Y – y = – (dx/dy)(X – x) ...(i)

It is given that 

Now, (dx/dy) = 0

x = c

which is passing through (1, 1), so c =1 

Hence, the equation of the curve is x = 1 

Also, dx/dy = 2xy/(x² – y²)

= (x² – y²)/2xy

log |1 + v²| = log c – log |x|

(1 + v²) = c/x

x² + y² = cx

which is passes through (1, 1), so c = 2

Hence, the equation of the curve is

x² + y² = 2x