Circle A has a radius of 3  and a center of (5 ,4 ). Circle B has a radius of 1  and a center of (7 ,2 ). If circle B is translated by <3 ,-5 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Feb 16, 2018

$\text{no overlap } , \approx 4.602$

Explanation:

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

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

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

$\text{before calculating d we require to find the centre of }$
$\text{B under the given translation}$

$\text{under the translation } < 3 , - 5 >$

$\left(7 , 2\right) \to \left(7 + 3 , 2 - 5\right) \to \left(10 , - 3\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

$\text{let "(x_1,y_1)=(5,4)" and } \left({x}_{2} , {y}_{2}\right) = \left(10 , - 3\right)$

$d = \sqrt{{\left(10 - 5\right)}^{2} + {\left(- 3 - 4\right)}^{2}} = \sqrt{25 + 49} \approx 8.062$

$\text{sum of radii } = 3 + 1 = 4$

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

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

$\textcolor{w h i t e}{\times \times \times \times \times} = 8.062 - 4 = 4.062$
graph{((x-5)^2+(y-4)^2-9)((x-10)^2+(y+3)^2-1)=0 [-20, 20, -10, 10]}