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

Apr 8, 2017

$\text{circles overlap}$

Explanation:

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

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

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

Before finding d we require to find the coordinates of the 'new' centre of B under the given translation which does not change the shape of the circle only it's position.

$\text{ Under a trans;ation } \left(\begin{matrix}2 \\ 1\end{matrix}\right)$

$\left(3 , 4\right) \to \left(3 + 2 , 4 + 1\right) \to \left(5 , 5\right) \leftarrow \textcolor{red}{\text{ new centre of B}}$

Since the x-ccordinates of both centres are 5 then the centres lie on the same vertical line and d is the difference in their y-coordinates.

$\Rightarrow d = 5 - 2 = 3$

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

$\text{since sum of radii"> d" then circles overlap}$
graph{(y^2-4y+x^2-10x+25)(y^2-10y+x^2-10x+25)=0 [-13.8, 13.91, -6.93, 6.92]}