question_answer

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

A) \[(y-2)y{{'}^{2}}=25-{{(y-2)}^{2}}\]

B) \[{{(y-2)}^{2}}y{{'}^{2}}=25-{{(y-2)}^{2}}\]

C) \[{{(x-2)}^{2}}y{{'}^{2}}=25-{{(y-2)}^{2}}\]

D) \[(x-2)y{{'}^{2}}=25-{{(y-2)}^{2}}\]

Correct Answer: B

Solution :

[b] Circle \[{{(x-h)}^{2}}+{{(y-2)}^{2}}=25\] \[\Rightarrow 2(x-h)+2(y-2)y'=0\Rightarrow 2(y-2)y'\] \[=-2(x-h)\] \[\Rightarrow {{(y-2)}^{2}}y{{'}^{2}}={{(x-h)}^{2}}\Rightarrow {{(y-2)}^{2}}y{{'}^{2}}=25-{{(y-2)}^{2}}\]