Question

The differential equation of the family of circles touching y-axis at the origin is:

• A. $(x^2+y^2)\frac{dy}{dx}-2xy=0$
• B. $x^2-y^2+2xy\frac{dy}{dx}=0$
• C. $(x^2-y^2)\frac{dy}{dx}-2xy=0$
• D. $(x^2+y^2)\frac{dy}{dx}+2xy=0$

Answer 1

The correct option is B $x^2-y^2+2xy\frac{dy}{dx}=0$

The system of circles touching Y axis at origin will have centres on X axis. Let (a,0) be the centre of a circle. Then the radius of the circle should be a units, since the circle should touch Y axis at origin.

Equation of a circle with centre at $(a,0)$ and radius $a$

$\left(x─a\right)²+\left(y─0\right)²=a²$

That is,

$x²+y²─2ax=0$ ─────► (1)

The above equation represents the family of circles touching Y axis at origin. Here 'a' is an arbitrary constant.
In order to find the differential equation of system of circles touching Y axis at origin, eliminate the the arbitrary constant from equation(1)
Differentiating equation(1) with respect to x,

$2x+2ydy/dx─2a=0$
or
$2a=2(x+ydy/dx)$

Replacing '2a' of equation(1) with the above expression, you get

$x²+y²─2(x+ydy/dx)\left(x\right)=0$

That is,
$─x²+y²─2xydy/dx=0$
or
$x²─y²+2xydy/dx=0$