Question

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

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

Answer 1

Answer: (A)

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

Equation of circle can be $(x-a{)}^2+(y-2{)}^2=25$ ...(1)
now differentiate the above eq with respect to x, we get

$2(x-a)+2(y-2)y^{\prime }=0$

$x-a+y-2=0$

$a=x+(y-2)y^{\prime }$ ....(2)

now substitute the value of a from eq(2) into eq(1), we get

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