Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1,2).

Explanation

The equation of the tangent at any point `P(x,y)` is
`Y-y=(dy)/(dx) (X-x)`
Given, that intercept on X-axis (putting Y=0)=2 (x-coordinates of P). Thus,
`x-y(dx)/(dy)=2x`
or `-(dy)/(dx)=(dx)/(x)`
Integrating we get, xy=c
Since the curve passes through `(1,2),c=-2`.
Hence, the equation of the required curve is `xy=-2`.