Form the differential equation of the family of parabolas with focus at the origin and the axis of symmetry along the axis.

Options

A. `y((dy)/(dx))^(2)+4x(dy)/(dx)=4y`
B. `y((dy)/(dx))^(2)=2x(dy)/(dx)-y`
C. `y((dy)/(dx))^(2)+y=2xy(dy)/(dx)`
D. `y((dy)/(dx))^(2)+2xy(dy)/(dx)+y-0`

Answer

B

Explanation

The equation of the family of parabolas with focus at the origin and the x-axis as axis is
`y^(2)=2a(x-1)`, where a is parameter.`" …(i)"`
Differentiating with respect to x, we get
`2y(dy)/(dx)=4arArr a=(y)/(2)(dy)/(dx)`
Sunstituting the value of a in (i), we get
`y^(2)=2y(dy)/(dx)(x-(y)/(2)(dy)/(dx))`
`rArr" "y^(2)=y(dy)/(dx)(2x-y(dy)/(dx))rArry((dy)/(dx))^(2)-2x(dy)/(dx)+y=0`
This is the required differential equation.