The number of common tangents that can be dra | vedclass

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(A) The given equations of the circles are $x^2+y^2-2x-2y-23=0 \dots (1)$ and $x^2+y^2-4x-4y-1=0 \dots (2)$.For circle $(1)$,the centre $C_1 = (1, 1)$

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(A) The given equations of the circles are $x^2+y^2-2x-2y-23=0 \dots (1)$ and $x^2+y^2-4x-4y-1=0 \dots (2)$.For circle $(1)$,the centre $C_1 = (1, 1)$ and radius $r_1 = \sqrt{1^2+1^2-(-23)} = \sqrt{25} = 5$.For circle $(2)$,the centre $C_2 = (2, 2)$ and radius $r_2 = \sqrt{2^2+2^2-(-1)} = \sqrt{9} = 3$.The distance between the centres $C_1$ and $C_2$ is $d = \sqrt{(2-1)^2+(2-1)^2} = \sqrt{1+1} = \sqrt{2} \approx 1.414$.We compare $d$ with the difference of the radii $|r_1 - r_2| = |5 - 3| = 2$.Since $d Therefore,the number of common tangents that can be drawn is $0$.

The number of common tangents that can be drawn to the circles $x^2+y^2-2x-2y-23=0$ and $x^2+y^2-4x-4y-1=0$ is

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