# Circle A has a center at (3 ,1 ) and an area of 15 pi. Circle B has a center at (5 ,2 ) and an area of 24 pi. Do the circles overlap?

Sep 7, 2016

Circles intersect each other and greatest possible distance between a point on one circle and another point on the other is $10.908$.

#### Explanation:

The center of first circle is $\left(3 , 1\right)$ and as area is $15 \pi$, the radius is $\sqrt{15} = 3.873$ (as $\pi {r}^{2} = 15 \pi$, $r = \sqrt{15}$) and center of second circle is $\left(5 , 2\right)$ and radius is $\sqrt{24} = 4.899$.

The distance between centers is $\sqrt{{\left(5 - 3\right)}^{2} + {\left(2 - 1\right)}^{2}}$

= $\sqrt{4 + 1} = \sqrt{5} = 2.236$

Let the radii of two circles is ${r}_{1}$ and ${r}_{2}$ and we also assume that ${r}_{1} > {r}_{2}$ and the distance between centers is $d$.

So here ${r}_{1} + {r}_{2} = 8.772 > d = 2.236$ and ${r}_{1} - {r}_{2} = 1.026 < d = 2.236$

Hence, the two circles intersect each other and greatest possible distance between a point on one circle and another point on the other is $3.873 + 4.899 + 2.236 = 10.908$

graph{(x^2+y^2-6x-2y-5)(x^2+y^2-10x-4y+5)=0 [-6.25, 13.75, -2.92, 7.08]}