A curve is 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 $$\left( {1,\,2} \right).$$  The equation of the curve is :

Solution :

Let $$P\left( {x,\,y} \right)$$  be any point on the curve, $$PM$$  the perpendicular to $$x$$-axis $$PT$$  the tangent at $$P$$ meeting the axis of $$x$$ at $$T.$$ As given $$OT = 2\,OM =2x.$$    Equation of the tangent at $$P\left( {x,\,y} \right)$$  is $$Y - y = \frac{{dy}}{{dx}}\left( {X - x} \right)$$
It intersects the axis of $$x$$ where $$Y = 0$$
i.e., $$ - y = \frac{{dy}}{{dx}}\left( {X - x} \right){\text{ or }}X = x - y\frac{{dy}}{{dx}} = OT$$
Hence, $$x - y\frac{{dy}}{{dx}} = 2x{\text{ or }}\frac{{dx}}{x} + \frac{{dy}}{y} = 0$$
Integrating, $$\log \,x + \log \,y = \log \,C{\text{ i}}{\text{.e}}{\text{., }}xy = C$$
This passes through $$\left( {1,\,2} \right),$$
$$\therefore \,C = 2$$
Hence the required curve is $$xy = 2$$