The differential equation of the family of parabolas with focus at the origin and the X-axis as axis, is

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= 2xy(dy)/(dx)+y=0`

Answer

b

Explanation

Equation of family of parabolas with focus at (0,0) and X = axis as axis is
` y^(2) = 4a (x+a)" "` (i) On differentiating Eq. (i) ,we get
` 2yy_(1) = 2a " " [ Here , y = (dy)/(dx)]`
On putting the value of a in Eq . (i)
` rArr y^(2) = 2 yy_(1) (x+ (yy_(1))/2)`
` rArr y ((dy)/dx)^(2) + 2x (dy)/(dx)=y`