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)

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