The curve whose sub tangent is ‘n’ times the abscissa of the point of contact and passes through the point (2,3), then

If $(x,y)$ is any point on the curve, the sub tangent at  $(x,y)=y\frac{dx}{dy}$
 $\therefore y\frac{dx}{dy}=nx\left(given\right)$ or   $n\frac{dy}{y}=\frac{dx}{x}$
Integrating  $n\log y=logx+\log c$ or  $\log y^n=\log cx$
or  $y^n=cx\mathrm{.......}\left(i\right)$
 which is the required equation of the family of curves.
Putting  $x=2,y=3$  in 
(i), we have  $3^n=2c$ or  $c=\frac{3^n}{2}$ Putting this value of c in 
(i)  $y^n=\frac{3^n}{2}x$  or   $2y^n=3^{n}x$
(ii) which is the particular curve passing through the point (2,3)
Putting  $n=1$ in
(ii), we have  $2y=3x$ which is a straight line putting  $n=2$  in
(ii) we have  $2y^2=9x$ which is a parabola.