Question

The differential equation of the family of circles touching the $Y$-axis at origin is

• A. $2y+x^2{y}^{\prime }=y^2$
• B. $2y{y}^{\prime }+x^2=y^2$
• C. $2x{y}^{\prime }+x^2=y^2$
• D. $2xy{y}^{\prime }+x^2=y^2$

Answer 1

Answer: (D)

The centre of the circle touching the $Y$-axis at origin lies on the $X$-axis.
Let $(a,0)$ be the centre of the circle and its radius is $a$.
Now, the equation of the family of circle with centre $(a,0)$ and radius a is
$(x-a{)}^2+y^2=a^2\Rightarrow x^2+y^2=2ax$
On differentiating Eq. (i) w.r.t. $x$, we get
$2x+2y{y}^{\prime }=2a\Rightarrow x+y{y}^{\prime }=a$
Now, substituting the value of a in Eq. (i), we get

$x^2+y^2=2\left(x+y{y}^{\prime }\right)x$
$\Rightarrow x^2+y^2=2x^2+2xy{y}^{\prime }$
$\Rightarrow 2xy{y}^{\prime }+x^2=y^2$
which is the required differential equation.