KCET · Maths · Circle
The number of common tangents to the circles \(\mathrm{x}^{2}+\mathrm{y}^{2}-\mathrm{y}=0\) and \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{y}=0\) is
- A 2
- B 3
- C 0
- D 1
Answer & Solution
Correct Answer
(B) 3
Step-by-step Solution
Detailed explanation
Given circles are \(\mathrm{x}^{2}+\mathrm{y}^{2}-\mathrm{y}=0\) and \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{y}=0\) centres and radii of these circles \(\operatorname{are} C_{1}\left(0, \frac{1}{2}\right), C_{2}\left(0,-\frac{1}{2}\right)\) and \(r_{1}=\frac{1}{2}, r_{2}=\frac{1}{2}\)
Now, \(\quad C_{1} C_{2}=\sqrt{0+\left(\frac{1}{2}+\frac{1}{2}\right)^{2}}=1\) and
\[
\because \quad \mathrm{C}_{1} \mathrm{C}_{2}=\mathrm{r}_{1}+\mathrm{r}_{2}
\]
It means that two circles touch each other externally.
Hence, number of common tangents are 3 .
Now, \(\quad C_{1} C_{2}=\sqrt{0+\left(\frac{1}{2}+\frac{1}{2}\right)^{2}}=1\) and
\[
\because \quad \mathrm{C}_{1} \mathrm{C}_{2}=\mathrm{r}_{1}+\mathrm{r}_{2}
\]
It means that two circles touch each other externally.
Hence, number of common tangents are 3 .
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