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 $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$ is

$
\left(\mathrm{x}-\mathrm{x}_1\right)\left(\mathrm{x}-\mathrm{x}_2\right)+\left(\mathrm{y}-\mathrm{y}_1\right)\left(\mathrm{y}-\mathrm{y}_2\right)+\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 $\mathrm{y}-\mathrm{y}_1=\mathrm{m}\left(\mathrm{x}-\mathrm{x}_1\right)$ at $\left(\mathrm{x}_1, \mathrm{y}_1\right)_{\text {for any finite }} \mathrm{m}$ is $\left(\mathrm{x}-\mathrm{x}_1\right)^2+\left(\mathrm{y}-\mathrm{y}_1\right)^2+\lambda\left\{\left(\mathrm{y}-\mathrm{y}_1\right)-\mathrm{m}\left(\mathrm{x}-\mathrm{x}_1\right)\right\}=0$ And if m is infinite then the family of circles is $\left(\mathrm{x}-\mathrm{x}_1\right)^2+\left(\mathrm{y}-\mathrm{y}_1\right)^2+\lambda\left(\mathrm{x}-\mathrm{x}_1\right)=0$