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# Circles

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Circles with centers of $(2,2)$ and $(17,17)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles?

Mar 17, 2024

#1
+394
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The equation of the first circle with center (2,2) is:

$${(x-2)}^{2}+{(y-2)}^{2}=4$$

The equation of the second circle with center (17, 17) is:

$${(x-17)}^{2}+{(y-17)}^{2}=289$$

Here is a graph:

The closest distance between the two is the length of the line segment conncting the 2 centers of the cricles minus the two radii.

Therefore the closest distance is $$\sqrt{(17-2)^2+(17-2)^2}-17-2=15\sqrt{2}-17-2$$.

Mar 18, 2024

#1
+394
+2

The equation of the first circle with center (2,2) is:

$${(x-2)}^{2}+{(y-2)}^{2}=4$$

The equation of the second circle with center (17, 17) is:

$${(x-17)}^{2}+{(y-17)}^{2}=289$$

Here is a graph:

The closest distance between the two is the length of the line segment conncting the 2 centers of the cricles minus the two radii.

Therefore the closest distance is $$\sqrt{(17-2)^2+(17-2)^2}-17-2=15\sqrt{2}-17-2$$.

hairyberry Mar 18, 2024