# Circle A has a center at (3 ,4 ) and an area of 64 pi. Circle B has a center at (1 ,12 ) and an area of 54 pi. Do the circles overlap?

May 7, 2018

The circles will intersect at two points.

#### Explanation:

Area of circle $A$ is ${A}_{A} = \pi \cdot {r}_{A}^{2} = 64 \pi \therefore {r}_{A} = 8$

Area of circle $B$ is ${A}_{B} = \pi \cdot {r}_{B}^{2} = 54 \pi$

$\therefore {r}_{B} = \sqrt{54} \approx 7.35 \left(2 \mathrm{dp}\right) \therefore {r}_{A} + {r}_{B} = 15.35$

and $| {r}_{A} - {r}_{B} | = 0.65$

Center of first circle $A$ is at $\left(3 , 4\right)$ and radius is $8$ unit .

Center of second circle $B$ is at $\left(1 , 12\right)$ and radius is

$\sqrt{54}$ unit . Distance between their centres is

$d = \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}} = \sqrt{{\left(3 - 1\right)}^{2} + {\left(4 - 12\right)}^{2}}$ or

$d = \sqrt{4 + 64} = \sqrt{68} \approx 8.25$ unit.

Two circles intersect if, and only if, the distance between their

centers is between the sum and the difference of their radii.

Here $0.65 < 8.25 < 15.35$, so they intersect at two points. [Ans]