Question

The differential equation satisfied by all the circle in the x - y plane is $(1+y_1^2)y_3=\lambda y_1{y}_2^2$ where $\lambda$ =

• A. 1
• B. 2
• C. -2
• D. 3

Answer 1

Answer: (D)

3

$x^2+y^2+2gx+2fy+c=0$

On differentiation:

$\Rightarrow 2x+2yy,+2g+2fy,=0$

On differentiation:

$\Rightarrow 2+2y_1^2+2y{y}_2+0+2f{y}_2=0$.......(1)

On differentiation:

$\Rightarrow 2y_1{y}_2+y_1{y}_2+y{y}_3+f{y}_3=0$

$\Rightarrow 3y_1{y}_2=y_3(-y-f)$.......(2)

From (1)

$(1+y_1^2)=-y_2(y+t)$

$\therefore -(y+t)=\frac{(1+y_1^2)}{y_2}$ .......(3)

Put (3) in (2)

$3y_1,y_2=y_3\left(\frac{(1+y_1^2)}{y_2}\right)$

$y_3(1+y_1^2)=3y_1{y}_2^2$

$\therefore \lambda =3$