Question

The differential equation of all circles which passes through the origin and whose centre lies on y-axis, is

• A. $\left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } - 2 x y = 0$
• B. $\left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } + 2 x y = 0$
• C. $\left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } - x y = 0$
• D. $\left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } + x y = 0$

Answer 1

Answer: (A)

$\left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } - 2 x y = 0$

Answer 2

The system of circles pass through origin and centre lies on y-axis is $x ^ { 2 } + y ^ { 2 } - 2 a y = 0$

⇒ $2 x + 2 y \frac { d y } { d x } - 2 a \frac { d y } { d x } = 0$ ⇒ $2 a = 2 y + 2 x \frac { d x } { d y }$

Therefore, the required differential equation is

$x ^ { 2 } + y ^ { 2 } - 2 y ^ { 2 } - 2 x y \frac { d x } { d y } = 0$ ⇒$\left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } - 2 x y = 0$.