Statement (A): The number of common tangents | vedclass

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(D) For circle $C_1: x^2 + y^2 = 4$,center $C_1 = (0, 0)$ and radius $r_1 = 2$.For circle $C_2: x^2 + y^2 - 6x - 8y - 24 = 0$,center $C_2 = (3, 4)$ an

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$A$ is false but $R$ is true.

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(D) For circle $C_1: x^2 + y^2 = 4$,center $C_1 = (0, 0)$ and radius $r_1 = 2$.For circle $C_2: x^2 + y^2 - 6x - 8y - 24 = 0$,center $C_2 = (3, 4)$ and radius $r_2 = \sqrt{3^2 + 4^2 - (-24)} = \sqrt{9 + 16 + 24} = \sqrt{49} = 7$.The distance between centers $|C_1C_2| = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = 5$.We check the condition $|C_1C_2|$ vs $r_1 + r_2$ and $|r_1 - r_2|$.$r_1 + r_2 = 2 + 7 = 9$.$|r_1 - r_2| = |2 - 7| = 5$.Since $|C_1C_2| = |r_1 - r_2| = 5$,the circles touch each other internally.When circles touch internally,they have only $1$ common tangent.Thus,Statement $(A)$ is false.Reason $(R)$ is a standard geometric theorem for circles,which is true.Therefore,$A$ is false but $R$ is true.

Statement $(A):$ The number of common tangents to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 6x - 8y = 24$ is $4$.
Reason $(R):$ For two circles with centers $C_1, C_2$ and radii $r_1, r_2$,if $|C_1C_2| > r_1 + r_2$,then the circles have $4$ common tangents.

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Statement $(A):$ The number of common tangents to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 6x - 8y = 24$ is $4$.Reason $(R):$ For two circles with centers $C_1, C_2$ and radii $r_1, r_2$,if $|C_1C_2| > r_1 + r_2$,then the circles have $4$ common tangents.