Question

The differential equation of the family of circles of fixed radius $r$ and having their centres on $y$-axis is

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

Answer 1

Answer: (A)

${\left(\frac{dy}{dx}\right)}^2=\frac{x^2}{r^2-x^2}$

Center is $(0,k)$ and radius is r.

Equation is

$x^2+(y-k{)}^2=r^2$-Equation 1

By differentiating we get,

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

$k=+x\frac{dx}{dy}+y$-Equation 2

Putting the value of K from Equation 1 & 2, we get

$x^2+(y-x\frac{dx}{dy}-y{)}^2=r^2$

$x^2+x^2(\frac{dx}{dy}{)}^2=r^2$

$1+(\frac{dx}{dy}{)}^2=\frac{r^2}{x^2}$

$(\frac{dy}{dx}{)}^2=\frac{x^2}{r^2-x^2}$