Question

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

• A. $(x-2)(y^1{)}^2=25-(y-2{)}^2$
• B. $(y-2)(y^1{)}^2=25+(y-2{)}^2$
• C. $(y-2{)}^2(y^1{)}^2=25-(y-2{)}^2$
• D. $(x-2{)}^2(y^1{)}^2=25-(y-2{)}^2$

Answer 1

Answer: (C)

$(y-2{)}^2(y^1{)}^2=25-(y-2{)}^2$

Let the general equation be

$(x-p{)}^2+(y-2{)}^2=25$-Eequation 1

By diifefrentiating , we get

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

$\to x-p=-(y-2)\frac{dy}{dx}$-Equation 2

From equation 1&2 we get,

$\to (y-2{)}^2[1+\left(\frac{dy}{dx}{)}^2\right]=25$

$\to (y-2{)}^2(\left(y^{{}^{\prime }}{)}^2\right)=25-(y-2{)}^2$