The number of common tangents to the circles | vedclass

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(B) The given circles are $x^{2}+y^{2}-y=0$ and $x^{2}+y^{2}+y=0$. For the first circle $x^{2}+y^{2}-y=0$,the center is $C_{1}(0, 1/2)$ and radius $r_

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$3$

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(B) The given circles are $x^{2}+y^{2}-y=0$ and $x^{2}+y^{2}+y=0$. For the first circle $x^{2}+y^{2}-y=0$,the center is $C_{1}(0, 1/2)$ and radius $r_{1} = \sqrt{0^{2} + (1/2)^{2} - 0} = 1/2$. For the second circle $x^{2}+y^{2}+y=0$,the center is $C_{2}(0, -1/2)$ and radius $r_{2} = \sqrt{0^{2} + (-1/2)^{2} - 0} = 1/2$. The distance between the centers is $C_{1}C_{2} = \sqrt{(0-0)^{2} + (1/2 - (-1/2))^{2}} = \sqrt{1^{2}} = 1$. Since $r_{1} + r_{2} = 1/2 + 1/2 = 1$,we have $C_{1}C_{2} = r_{1} + r_{2}$. This condition implies that the two circles touch each other externally at a single point. When two circles touch each other externally,they have exactly $3$ common tangents (two external and one internal).

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

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