The number of common tangents to the circles | vedclass

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

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(A) For the first circle $x^2 + y^2 - 2x - 1 = 0$,the center $C_1 = (1, 0)$ and radius $r_1 = \sqrt{1^2 + 0^2 - (-1)} = \sqrt{2}$.For the second circle $x^2 + y^2 - 2y - 7 = 0$,the center $C_2 = (0, 1)$ and radius $r_2 = \sqrt{0^2 + 1^2 - (-7)} = \sqrt{8} = 2\sqrt{2}$.The distance between the centers $C_1$ and $C_2$ is $d = \sqrt{(1-0)^2 + (0-1)^2} = \sqrt{1 + 1} = \sqrt{2}$.We observe that $|r_2 - r_1| = |2\sqrt{2} - \sqrt{2}| = \sqrt{2}$.Since $d = |r_2 - r_1|$,the two circles touch each other internally.When two circles touch internally,there is exactly $1$ common tangent.

The number of common tangents to the circles $x^2 + y^2 - 2x - 1 = 0$ and $x^2 + y^2 - 2y - 7 = 0$ is:

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