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

Feb 23, 2018

They do not overlap and the minimum distance between the two circles is >1.

#### Explanation:

To figure this out, the simplest thing to do is to graph it out. From all the information given, we can find the equations for both circles and graph them. A circle's equation is:

${x}^{2} + {y}^{2} = {r}^{2}$

$r$ in this case equals the $r a \mathrm{di} u s$ of the circle.

Also, the problem gives us the center point at which the circle sits. This will also help with forming the equations.

Circle A is ${\left(x - 2\right)}^{2} + {\left(y - 7\right)}^{2} = 25$
Circle B is ${\left(x - 3\right)}^{2} + {\left(y - 1\right)}^{2} = 1$

Now that we have our equations, we can graph them. graph{(x-2)^2+(y-7)^2=25 [-13.39, 14.7, 0.26, 14.31]}

graph{(x-3)^2+(y-1)^2=1 [-11.13, 13.84, -1.94, 10.55]}

Now, let's visualize Circle B translated to the right one, and up three. This will make Circle B's new equation look like:

${\left(x - 4\right)}^{2} + {\left(y - 10\right)}^{2} = 1$
graph{(x-4)^2+(y-10)^2=1 [-6.25, 13.48, 3.71, 13.58]}

If we now compare the graph of Circle A with the translated graph of Circle B, we can see that they still do not overlap and the minimum distance between the two circles is >1.

Feb 23, 2018

$\text{circle B is inside circle A}$

#### Explanation:

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

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

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

• " if difference of radii">d" then 1 circle inside other"

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

$\text{under a translation } < 1 , 3 >$

$\left(3 , 1\right) \to \left(3 + 1 , 1 + 3\right) \to \left(4 , 4\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

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

$\text{let "(x_1,y_1)=2,7)" and } \left({x}_{2} , {y}_{2}\right) = \left(4 , 4\right)$

$d = \sqrt{{\left(4 - 2\right)}^{2} + {\left(4 - 7\right)}^{2}} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61$

$\text{sum of radii } = 5 + 1 = 6$

$\text{difference of radii } = 5 - 1 = 4$

$\text{Since diff. of radii">d" then 1 circle inside other}$
graph{((x-2)^2+(y-7)^2-25)((x-4)^2+(y-4)^2-1)=0 [-40, 40, -20, 20]}