The number of common tangents to the circles | vedclass

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(C) For the circle $x^2+y^2-x=0$,the center $C_1 = (\frac{1}{2}, 0)$ and radius $r_1 = \sqrt{(\frac{1}{2})^2 + 0^2 - 0} = \frac{1}{2}$.For the circle

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$3$

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(C) For the circle $x^2+y^2-x=0$,the center $C_1 = (\frac{1}{2}, 0)$ and radius $r_1 = \sqrt{(\frac{1}{2})^2 + 0^2 - 0} = \frac{1}{2}$.For the circle $x^2+y^2+x=0$,the center $C_2 = (-\frac{1}{2}, 0)$ and radius $r_2 = \sqrt{(-\frac{1}{2})^2 + 0^2 - 0} = \frac{1}{2}$.The distance between the centers is $C_1C_2 = \sqrt{(\frac{1}{2} - (-\frac{1}{2}))^2 + (0-0)^2} = \sqrt{1^2} = 1$.The sum of the radii is $r_1 + r_2 = \frac{1}{2} + \frac{1}{2} = 1$.Since $C_1C_2 = r_1 + r_2$,the circles touch each other externally.When two circles touch each other externally,the number of common tangents is $3$.

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

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