The differential equation of all circles passing through the origin and having their centres on the x-axis is (1) `x^2=""y^2+""x y(dy)/(dx)` (2) `x^2=""y^2+"3"x y(dy)/(dx)` (3) `y^2=x^2""+"2"x y(dy)/(dx)` (4) `y^2=x^2""-"2"x y(dy)/(dx)`

Options

A. `y^(2)=x^(2)+2xy(dy)/(dx)`
B. `y^(2)=x^(2)-2xy(dy)/(dx)`
C. `x^(2)=y^(2)+xy(dy)/(dx)`
D. `x^(2)=y^(2)+3xy(dy)/(dx)`

Answer

A

Explanation

The equation of the family of circles passing through the origin and having their centres on x-axis is
`(x-a)^(2)+(y-0)^(2)=a^(2)or, x^(2)+y^(2)-2ax=0" …(i)"`
Differentiating w.r.to x, we get
`2x+2y(dy)/(dx)-2a=0rArr a= x+y(dy)/(dx)`
Substituting the value of a in (i), we get
`x^(2)+y^(2)-2x^(2)-2xy(dy)/(dx)=0 or, y^(2)=x^(2)+2xy(dy)/(dx)`
as the required differential equation.