The number of common tangents to the circles $x^2+y^2-x=0$ and $x^2+y^2+x=0$ is /are

The number of common tangents to the circles $x^2+y^2-x=0$ and $x^2+y^2+x=0$ is /are
  1. 1
  2. 2
  3. 3
  4. 4

Solution

$\begin{aligned} & x^2+y^2-x=0 \\ & \mathrm{C}_1=\left(\frac{1}{2}, 0\right), \mathrm{r}_1=\sqrt{\left(-\frac{1}{2}\right)^2+0^2-0}=\frac{1}{2} \\ & x^2+y^2+x=0 \\ & \mathrm{C}_2=\left(-\frac{1}{2}, 0\right), \mathrm{r}_2=\sqrt{\left(\frac{1}{2}\right)^2+0^2-0}=\frac{1}{2} \\ & \mathrm{C}_1 \mathrm{C}_2=\sqrt{\left(\frac{1}{2}+\frac{1}{2}\right)^2+0^2}=1 \\ & \mathrm{r}_1+\mathrm{r}_2=\frac{1}{2}+\frac{1}{2}=1 \\ & \mathrm{C}_1 \mathrm{C}_2=\mathrm{r}_1 \mathrm{r}_2 \\ & \Rightarrow \text { The given circles touch each other externally. } \\ & \Rightarrow \text { Number of common tangents }=3\end{aligned}$

Asked in: MHT CET 2024 (15 May Shift 1)

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