# Circle A has a center at (2 ,12 ) and an area of 81 pi. Circle B has a center at (1 ,3 ) and an area of 16 pi. Do the circles overlap? If not, what is the shortest distance between them?

Jul 30, 2016

circles overlap

#### Explanation:

What we have to do here is compare the distance ( d) between the centres of the circles to the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

Before doing this, we require to find the radii of both circles.

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder}}$ The area (A) of a circle is

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{A = \pi {r}^{2}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\textcolor{b l u e}{\text{Circle A }} \pi {r}^{2} = 81 \pi \Rightarrow {r}^{2} = \frac{81 \cancel{\pi}}{\cancel{\pi}} \Rightarrow r = 9$

$\textcolor{b l u e}{\text{Circle B }} \pi {r}^{2} = 16 \pi \Rightarrow {r}^{2} = \frac{16 \cancel{\pi}}{\cancel{\pi}} \Rightarrow r = 4$

To calculate d, use the $\textcolor{b l u e}{\text{distance formula}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where$\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

The 2 points here are (2 ,12) and (1 ,3) the centres of the circles.

let $\left({x}_{1} , {y}_{1}\right) = \left(2 , 12\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(1 , 3\right)$

d=sqrt((1-2)^2+(3-12)^2)=sqrt(1+81)=sqrt82≈9.055

sum of radii = radius of A + radius of B = 9 + 4 = 13

Since sum of radii > d , then circles overlap
graph{(y^2-24y+x^2-4x+67)(y^2-6y+x^2-2x-6)=0 [-56.96, 56.94, -28.5, 28.46]}