Statement (A) : If two circles x^2 + y^2 + 2g | vedclass

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(C) For Statement $(A) :$ Two circles $x^2 + y^2 + 2gx + 2fy = 0$ and $x^2 + y^2 + 2g'x + 2f'y = 0$ touch each other if the distance between their cen

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

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(C) For Statement $(A) :$ Two circles $x^2 + y^2 + 2gx + 2fy = 0$ and $x^2 + y^2 + 2g'x + 2f'y = 0$ touch each other if the distance between their centers $(C_1, C_2)$ is equal to the sum or difference of their radii $(r_1 \pm r_2)$.Centers are $C_1(-g, -f)$ and $C_2(-g', -f')$. Radii are $r_1 = \sqrt{g^2 + f^2}$ and $r_2 = \sqrt{g'^2 + f'^2}$.The condition is $C_1C_2^2 = (r_1 \pm r_2)^2$.$(-g + g')^2 + (-f + f')^2 = g^2 + f^2 + g'^2 + f'^2 \pm 2\sqrt{g^2 + f^2}\sqrt{g'^2 + f'^2}$.$g^2 - 2gg' + g'^2 + f^2 - 2ff' + f'^2 = g^2 + f^2 + g'^2 + f'^2 \pm 2\sqrt{g^2 + f^2}\sqrt{g'^2 + f'^2}$.$-2(gg' + ff') = \pm 2\sqrt{g^2 + f^2}\sqrt{g'^2 + f'^2}$.Squaring both sides: $(gg' + ff')^2 = (g^2 + f^2)(g'^2 + f'^2)$.$g^2g'^2 + f^2f'^2 + 2gg'ff' = g^2g'^2 + g^2f'^2 + f^2g'^2 + f^2f'^2$.$2gg'ff' = g^2f'^2 + f^2g'^2$.$g^2f'^2 + f^2g'^2 - 2gg'ff' = 0$.$(gf' - fg')^2 = 0 \Rightarrow gf' = fg'$.Thus,Statement $(A)$ is true.For Statement $(R) :$ The line joining the centers of two touching circles is perpendicular to the common tangent at the point of contact,not to all common tangents. Thus,Statement $(R)$ is false.

Statement $(A) :$ If two circles $x^2 + y^2 + 2gx + 2fy = 0$ and $x^2 + y^2 + 2g'x + 2f'y = 0$ touch each other,then $f'g = fg'$.
Reason $(R) :$ Two circles touch each other if the line joining their centers is perpendicular to all possible common tangents.

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Statement $(A) :$ If two circles $x^2 + y^2 + 2gx + 2fy = 0$ and $x^2 + y^2 + 2g'x + 2f'y = 0$ touch each other,then $f'g = fg'$.Reason $(R) :$ Two circles touch each other if the line joining their centers is perpendicular to all possible common tangents.