Question
Under which one of the following conditions does the solution of $$\frac{{dy}}{{dx}} = \frac{{ax + b}}{{cy + d}}$$ represent a parabola ?
A.
$$a = 0,\,c = 0$$
B.
$$a = 1,\,b = 2,\,c \ne 0$$
C.
$$a = 0,\,c \ne 0,\,b \ne 0$$
D.
$$a = 1,\,c = 1$$
Answer :
$$a = 0,\,c \ne 0,\,b \ne 0$$
Solution :
Given : $$\frac{{dy}}{{dx}} = \frac{{ax + b}}{{cy + d}}$$
or, $$\left( {cy + d} \right)dy = \left( {ax + b} \right)dx$$
Integrating both the sides.
$$\eqalign{
& c.\int {y\,dy + d} \int {dy = a} \int {x\,dx + b} \int {dx + K} \,\,\,\,\,\,\left[ {K{\text{ is constant integration}}} \right] \cr
& {\text{or, }}c.\frac{{{y^2}}}{2} + d.y = a\frac{{{x^2}}}{2} + b.x + K \cr
& {\text{or, }}c{y^2} + 2d.y = a{x^2} + 2b.x + 2K \cr} $$
This equation will represent a parabola when either, the coefficient of $${x^2}$$ or the coefficient of $${y^2}$$ is zero, but not both.
Thus either $$c = 0$$ or $$a = 0$$ but not both.
From the choice given $$a = 0,\,c \ne 0{\text{ and }}\,b \ne 0$$