Question

Consider the family of all circles having their centre at the point $(1,2)$. Write the equation of the family.
Write the corresponding differential equation.

Answer 1

The equation of the family of circles having the centre at the point $(x_1,y_1)$ is given by

${(x-x_1)}^2+{(y-y_1)}^2=r^2$

$Here(x_1,y_1)=(1,2)$

$\therefore$ the equation is given by ${(x-1)}^2+{(y-2)}^2=r^2$

Differentiating the above equation

$\Rightarrow$ $2(x-1)+2(y-2)\frac{dy}{dx}=0$

$\Rightarrow$ $\frac{dy}{dx}=\frac{(1-x)}{(y-2)}$