# Circle A has a center at (4 ,-1 ) and a radius of 5 . Circle B has a center at (-3 ,6 ) and a radius of 2 . Do the circles overlap? If not, what is the smallest distance between them?

Jun 6, 2017

They don't overlap. The closest they get is $7 \sqrt{2} - 7$.

#### Explanation:

The radii of the two circles are $2$ and $5$, so the circles will touch or overlap if the distance between the centres is $7$ or less.
We can work out the distance between the centres with Pythagoras. The distance in x is $4 - - 3 = 7$, and the distance in y is $6 - - 1 = 7$, so the distance between the two centres is $\sqrt{{7}^{2} + {7}^{2}} = 7 \sqrt{2}$. As the distance between the two centres is more than the sum of the two radii, they cannot touch.
The closest distance, therefore, will be $7 \sqrt{2} - 7$, the distance between the centres minus the radii.

Jun 6, 2017

$\text{no overlap} , \approx 2.899$

#### Explanation:

$\text{what we have to do here is "color(blue)"compare ""the distance (d)}$
$\text{between the centres of the circles to the "color(blue)"sum of radii}$

• " if sum of radii" > d" then circles overlap"

• " if sum of radii"< d" then no overlap"

$\text{to calculate d use the "color(blue)"distance formula}$

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

$\text{the 2 points are } \left({x}_{1} , {y}_{1}\right) = \left(4 , - 1\right) , \left({x}_{2} , {y}_{2}\right) = \left(- 3 , 6\right)$

$d = \sqrt{{\left(- 3 - 4\right)}^{2} + {\left(6 + 1\right)}^{2}} = \sqrt{49 + 49} = \sqrt{98} \approx 9.899$

$\text{sum of radii } = 5 + 2 = 7$

$\text{Since sum of radii"< d" then no overlap}$

$\text{smallest distance "=d-" sum of radii}$

$\textcolor{w h i t e}{s m a l \le s t \mathrm{di} s \tan c e} = 9.899 - 7$

$\textcolor{w h i t e}{s m a l \le s t \mathrm{di} s \tan c e} = 2.899$
graph{(y^2+2y+x^2-8x-8)(y^2-12y+x^2+6x+41)=0 [-20, 20, -10, 10]}