Question

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

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

Answer 1

The correct option is B ${(y-2)}^2{y}^{\prime 2}=25-{(y-2)}^2$

Given centre lies on the line $y=2$
$\therefore$ $C(\alpha ,2)$
and radius of circle$=5$ units
$\therefore$ equation of circle be
${(x-\alpha )}^2+{(y-2)}^2=25$ (A)
Differentiating (A) w.r. to x we get
$(x-\alpha )+(y-2)y^{\prime }=0$
$\Rightarrow$ ${(x-\alpha )}^2={(y-2)}^2{y}^{\prime 2}$ (B)
Substitute (B) in (A), we have
${(y-2)}^2{\left(y^{\prime }\right)}^2+{(y-2)}^2=25$
$\Rightarrow$ ${(y-2)}^2{\left(y^{\prime }\right)}^2=25-{(y-2)}^2$