Which one of the following differential equation represents the system of circles touching y-axis at the origin ?

Options

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

Answer

B

Explanation

Since, circle is touching y-axis at origin its centre lies on x-axis . Let the centre be (a,0) . Its radius =a
` (x -a)^(2) +y^(2) =a^(2)`
` x^(2) +a^(2) -2ax +y^(2) =a^(2)`
` Rightarrow x^(2) +y^(2) -2ax =0`
`a = (x^(2) +y^(2))/(2x)`
Differenting both sides
Now, ` 2x+2y (dy)/(dx) -2a =0`
` Rightarrow 2x +2y (dy)/(dx)- (x^(2) +y^(2))/x =0`
` Rightarrow 2xy(dy)/(dx) =y^(2) -y^(2) =0`