Question

The circles $x^2 + y^ 2 + 6x + 6y = 0$ and $x^2 + y ^2 - 12x - 12y = 0$

• A. cut orthogonally
• B. touch each other internally
• C. intersect in two points
• D. touch each other externally

Answer 1

Answer: (D)

Given equation of circles are

$x^2 + y^ 2 + 6x + 6y = 0 ...(i)$

and $x^2 + y^ 2 - 12x - 12y = 0 ...(ii)$

Here, $g_1 = 3, f _1 = 3, g_ 2 = - 6$ and $f _2 = - 6$

$\therefore $ Centres of circles are $C_1 (-3, - 3)$ and $C_2(6, 6)$

respectively and radii are $r_1 = 3 \sqrt 2$ and $r_2 = 6\sqrt 2$
respectively.

Now, $C_{1}C_{2} = \sqrt{\left(6+3\right)^{2}+\left(6+3\right)^{2}} $

$= 9\sqrt{2} $

and $r_{1}+r_{2} = 3 \sqrt{2} +6\sqrt{2} = 9\sqrt{2} $

$ \Rightarrow C_{1}C_{2} = r_{1} + r_{2}$

$\therefore $ Both circles touch each other externally.