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

Options

A. `(y-2)^(2)y'^(2)=25-(y-2)^(2)`
B. `(x-2)^(2)y'^(2)=25-(y-2)^(2)`
C. `(x-2)y'^(2)=25-(y-2)^(2)`
D. `(y-2)y'^(2)=25-(y-2)^(2)`

Answer

A

Explanation

Let (a, 2) be the centre of the circle, where 'a' is a variable. Then, the equation of the family of circles is
`(x-a)^(2)+2(y-2)^(2)=5^(2)" …(i)"`
Differentiating w.r. to x, we get
`2(x-a)+2(y-2)(dy)/(dx)rArr x-a = -(y-2)y_(1)`
Substituting this value of `(x-a)` in (i), we get
`(y-2)^(2)y_(1)^(2)=25-(y-2)^(2)`
as the requried differential equation.