Question

The differential equation of the family of circles with fixed radius $5$ units and centre on the line $y=2$ is

• A. $(x-2)y^{\prime 2}=25-(y-2{)}^2$
• B. $(y-2)y^{\prime 2}=25-(y-2{)}^2$
• C. $(y-2{)}^2{y}^{\prime 2}=25-(y-2{)}^2$
• D. $(x-2{)}^2{y}^{\prime 2}=25-(y-2{)}^2$

Answer 1

Answer: (C)

$(x-h{)}^2+(y-2{)}^2=\mathrm{25....}$ (1)
$\Rightarrow 2(x-h)+2(y-2)\frac{dy}{dx}=0$
$\Rightarrow (x-h)=-(y-2)\frac{dy}{dx}$
Substituting in Equation (1), we have
$(y-2{)}^2{\left(\frac{dy}{dx}\right)}^2+(y-2{)}^2=25$
$(y-2{)}^2{y}^{\prime 2}=25-(y-2{)}^2$