Under which one of the following conditions does the solution of `(dy)/(dx)=(ax +b)/(cy +d)` represent a parabola ?

Options

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

C

Explanation

Given : `(dy)/(dx) = (ax +b)/(cy +d)`
or (cy +d) dy = (ax +b) dx
Integrating both the sides.
` c.intydy+dintdy = a int xd x + b int dx +K`
` or, c.y^(2)/2 = d.y =a x^(2)/2 + b.x +K`
`or ,cy^(2) +2d.y = ax^(2) +2b.x +2K`
This is 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 choise given , a=0 , ` cne 0 and b ne 0`