# Circle A has a center at (8 ,5 ) and an area of 100 pi. Circle B has a center at (4 ,2 ) and an area of 36 pi. Do the circles overlap? If not, what is the shortest distance between them?

Jul 31, 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 the circles.

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

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

$\textcolor{b l u e}{\text{Circle A }} \pi {r}^{2} = 100 \pi \Rightarrow {r}^{2} = \frac{100 \cancel{\pi}}{\cancel{\pi}} \Rightarrow r = 10$

$\textcolor{b l u e}{\text{Circle B }} \pi {r}^{2} = 36 \pi \Rightarrow {r}^{2} = \frac{36 \cancel{\pi}}{\cancel{\pi}} \Rightarrow r = 6$

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 (8 ,5) and (4 ,2) the centres of the circles.

$d = \sqrt{{\left(8 - 4\right)}^{2} + {\left(5 - 2\right)}^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5$

sum of radii = radius of A + radius of B = 10 + 6 = 16

Since sum of radii > d , then circles overlap
graph{(y^2-4y+x^2-8x-16)(y^2-10y+x^2-16x-11)=0 [-9.86, 9.88, -4.93, 4.935]}