Family of Circles - Practice Questions & MCQ

Family of Circles

1. Equation of the family of circles passing through the point of intersection of two given circles $S=0$ and $S^{\prime }=0$ is $S+\lambda S^{\prime }=0$ where $\lambda$ is the parameter

2. Equation of the family of circles passing through the point of intersection of a given circle $S=0$ and a line $L=0$ is $S+\lambda L=0$ where $\lambda$ is the parameter.

3. Equation of the family of circles touching the given circle $S=0$ and the line $L=0$ is $S+\lambda L=0$

4. The equation of the family of circles passing through the two given points $P(x_1,y_1)$ and $Q(x_2,y_2)$ is

$(\mathrm{x}-{\mathrm{x}}_1)(\mathrm{x}-{\mathrm{x}}_2)+(\mathrm{y}-{\mathrm{y}}_1)(\mathrm{y}-{\mathrm{y}}_2)+\lambda \left|\begin{array}{lll}x& y& 1\\ x_1& y_1& 1\\ x_2& y_2& 1\end{array}\right|=0$

5. The equation of the family of circles which touch $\mathrm{y}-{\mathrm{y}}_1=\mathrm{m}(\mathrm{x}-{\mathrm{x}}_1)$ at ${({\mathrm{x}}_1,{\mathrm{y}}_1)}_{\text{for any finite }}\mathrm{m}$ is ${(\mathrm{x}-{\mathrm{x}}_1)}^2+{(\mathrm{y}-{\mathrm{y}}_1)}^2+\lambda \{(\mathrm{y}-{\mathrm{y}}_1)-\mathrm{m}(\mathrm{x}-{\mathrm{x}}_1)\}=0$ And if m is infinite then the family of circles is ${(\mathrm{x}-{\mathrm{x}}_1)}^2+{(\mathrm{y}-{\mathrm{y}}_1)}^2+\lambda (\mathrm{x}-{\mathrm{x}}_1)=0$