Animated Solution for Mathematics - Circles: Choose the correct statement about two circles whose equations are given below: $x^2+y^2-10x-10y+41=0$ and $x^2+y^2-22x-10y+137=0$

To understand the interaction between the two circles, we must first identify their fundamental properties: their centers and radii. The general equation of a circle is given by:

For a circle in this form, the center is $(-g,-f)$ and the radius is calculated as $R=\sqrt{g^2+f^2-c}$.

For the first circle, $C_1$, given by $x^2+y^2-10x-10y+41=0$, we identify $g=-5$, $f=-5$, and $c=41$.

The center $A$ is $(5,5)$. The radius $R_1$ is:

For the second circle, $C_2$, given by $x^2+y^2-22x-10y+137=0$, we identify $g=-11$, $f=-5$, and $c=137$.

The center $B$ is $(11,5)$. The radius $R_2$ is:

Now that we have the centers $A(5,5)$ and $B(11,5)$, we calculate the distance $d$ between them. Since the $y$-coordinates are identical, the distance is simply the difference in the $x$-coordinates:

We compare the distance between the centers to the sum of the radii. We have $R_1=3$ and $R_2=3$, so:

Since the distance between the centers $d$ is exactly equal to the sum of the radii ($d=R_1+R_2$), the circles are in a state of perfect external tangency.

They touch at exactly one point. The circles do not overlap, nor are they separated by any distance.