Question

The order and degree of the differential equation of the family of the circles touching the $x$-axis at the origin, are respectively:

• A. $1,1$
• B. $1,2$
• C. $2,1$
• D. $2,2$

Answer 1

The correct option is A $1,1$

The system of circles touching $x$-axis at origin will have centres on $y$-axis.

Let $(0,a)$ be the centre of a circle.

Then the radius of the circle should be $a$ units, since the circle should touch $x$-axis at origin.
Equation of a circle with centre at $(0,a)$ and radius $a$ is
$(x-0{)}^2+(y-a{)}^2=a^2$
$\Rightarrow x^2+y^2-2ay=0$
The above equation represents the family of circles touching $x$-axis at origin. Here '$a$' is an arbitrary constant.
In order to find the differential equation of system of circles touching $x$-axis at origin, eliminate the the arbitrary constant from equation
Differentiating equation with respect to $y$,
$2x\frac{dx}{dy}+2y-2a=0$
$\frac{dx}{dy}=2(y-a)$
$\Rightarrow$ Order is $1$ and degree is also $1$.
Hence, option 'A' is correct.