# Circle A has a center at (1 ,4 ) and an area of 100 pi. Circle B has a center at (7 ,9 ) and an area of 36 pi. Do the circles overlap? If not, what is the shortest distance between them?

Nov 30, 2016

The two circles intersect each other and hence overlap. The shortest distance between them is zero (at the points they intersect).

#### Explanation:

Whether the two circles with given centers and radii intersect each other, touch each other (internally or externally) are separated and do not overlap or bigger circle engulfs smaller circle, critically depends on the distance between the centers and sum / difference of their radii . One can find the details here.

Here the distance between centers $\left(1 , 4\right)$ and $\left(7 , 9\right)$ is $\sqrt{{\left(7 - 1\right)}^{2} + {\left(9 - 4\right)}^{2}} = \sqrt{36 + 25} = \sqrt{61} = 7.8102$.

As areas of circle are given by $\pi {r}^{2}$, radius is $\sqrt{\frac{A r e a}{\pi}}$ and while radius of first circle with center $\left(1 , 4\right)$ is $\sqrt{\frac{100 \pi}{\pi}} = \sqrt{100} = 10$ and radius of second circle with center $\left(7 , 9\right)$ is $\sqrt{\frac{36 \pi}{\pi}} = \sqrt{36} = 6$.

Sum of radii is $16$ and difference of radii is $4$.

As sum of radii is greater than distance between centers and difference between radii is less than distance between centers, the two circles intersect each other and hence overlap. The shortest distance between them is zero (at the points they intersect).
graph{(x^2+y^2-2x-8y-75)(x^2+y^2-14x-18y+94)=0 [-17.17, 22.83, -4.64, 15.36]}