Question

Find the differential equation of the family of circles of fixed radius r and having their centers on the x-axis ?

Answer 1

Hello,

As center lies on x -axis .

let the coordinates of circle are ( a,0)

radius is r

general equation of circle is

$( {x - a)}^{2} + ( {y - b)}^{2} = {r}^{2}$
$( {x - a})^{2} + {y}^{2} = {r}^{2}$
------eq1

differentiate eq1 with respect to x

$2(x - a) + 2y \times \frac{dy}{dx} = 0 \\ \\ x - a + y \frac{dy}{dx} = 0 \\ \\ - y \frac{dy}{dx} = (x - a)$
put the value of (x-a) in eq 1

$( { - y \frac{dy}{dx}) }^{2} + {y}^{2} = {r}^{2} \\ \\ {y}^{2} ( { \frac{dy}{dx} )}^{2} + {y}^{2} - {r}^{2} = 0$
is the required differential equation.

Answer 2

Question: The D.E. of a family of circles having radius ''r'' and the center on the X-axis .??