When do the two circles x^2 + y^2 = ax and x^ | vedclass

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(D) For the first circle $x^2 + y^2 - ax = 0$,the center is $C_1 = (a/2, 0)$ and the radius is $r_1 = |a/2|$.For the second circle $x^2 + y^2 = c^2$,t

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$|a| = c$

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(D) For the first circle $x^2 + y^2 - ax = 0$,the center is $C_1 = (a/2, 0)$ and the radius is $r_1 = |a/2|$.For the second circle $x^2 + y^2 = c^2$,the center is $C_2 = (0, 0)$ and the radius is $r_2 = c$.The two circles touch each other if the distance between their centers is equal to the sum or difference of their radii: $d = |r_1 \pm r_2|$.The distance between centers $d = \sqrt{(a/2 - 0)^2 + (0 - 0)^2} = |a/2|$.Setting $d = r_1 + r_2$ gives $|a/2| = |a/2| + c$,which implies $c = 0$ (not possible as $c > 0$).Setting $d = |r_1 - r_2|$ gives $|a/2| = | |a/2| - c |$.Squaring both sides: $(a/2)^2 = (|a/2| - c)^2$.$(a/2)^2 = (a/2)^2 - 2|a/2|c + c^2$.$0 = -|a|c + c^2$.$|a|c = c^2$.Since $c > 0$,we divide by $c$ to get $|a| = c$.

When do the two circles $x^2 + y^2 = ax$ and $x^2 + y^2 = c^2$ $(c > 0)$ touch each other?

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