Question

If a curve passing through P(4, 2) and satisfies the differential equation $\frac{dy}{dx}=\frac{y}{x-y^2}$ is a conic then which of the following is correct?

• A. focus of conic lies on line y = 2
• B. equation of directrix of conic is 4x – 17 = 0
• C. eccentricity of conic is 1
• D. eccentricity of conic is 3

Answer 1

Answer: (A), (B), (C)

The correct options are
A

focus of conic lies on line y = 2

B

equation of directrix of conic is 4x – 17 = 0

C

eccentricity of conic is 1

$xdy-y^{2}dy=ydx\Rightarrow -y^{2}dy=ydx-xdy\Rightarrow -dy=d\left(\frac{x}{y}\right)-y=\frac{x}{y}+C-2=\frac{4}{2}+C\Rightarrow C=-4\therefore -y^2=x-4yx=4y–y^2=4-(y-2{)}^2(y-2{)}^2=–(x–4)\therefore Focus\equiv (\frac{15}{4},2)$
Equation of directrix
$x=\frac{17}{4}\Rightarrow 4x-17=0e=1$