Circle A has a radius of 1  and a center of (2 ,4 ). Circle B has a radius of 2  and a center of (4 ,9 ). If circle B is translated by <1 ,-4 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jun 23, 2017

$\text{no overlap} , \approx 0.162$

Explanation:

What we have to do here is $\textcolor{b l u e}{\text{compare}}$ the distance ( d) between the centres to the $\textcolor{b l u e}{\text{sum of the radii}}$

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

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

$\text{before calculating d we require to find the new centre }$
$\text{of B under the given translation which does not change}$
$\text{the shape of the circle only it's position}$

$\text{under a translation } \left(\begin{matrix}1 \\ - 4\end{matrix}\right)$

$\left(4 , 9\right) \to \left(4 + 1 , 9 - 4\right) \to \left(5 , 5\right) \leftarrow \textcolor{red}{\text{ new centre of B}}$

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

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

$\text{the points are } \left({x}_{1} , {y}_{1}\right) = \left(2 , 4\right) , \left({x}_{2} , {y}_{2}\right) = \left(5 , 5\right)$

$d = \sqrt{{\left(5 - 2\right)}^{2} + {\left(5 - 4\right)}^{2}} = \sqrt{10} \approx 3.162$

$\text{sum of radii } = 1 + 2 = 3$

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

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

$\Rightarrow \text{minimum distance } = 3.162 - 3 = 0.162$
graph{(y^2-8y+x^2-4x+19)(y^2-10y+x^2-10x+46)=0 [-10, 10, -5, 5]}