# Circle A has a center at (2 ,3 ) and an area of 18 pi. Circle B has a center at (7 ,2 ) and an area of 64 pi. Do the circles overlap?

Mar 2, 2018

$d$ lies between $\left({R}_{B} - {R}_{A}\right) \mathmr{and} \left({R}_{B} + {R}_{A}\right)$, so circles
will overlap.

#### Explanation:

Circle "A" area is ${A}_{A} = \pi \cdot {R}_{A}^{2} = 18 \pi \therefore {R}_{A}^{2} = 18$

$\therefore {R}_{A} = \sqrt{18} \approx 4.24$ Centre :$\left({x}_{1} = 2 , {y}_{1} = 3\right)$

Circle "B" area is ${A}_{B} = \pi \cdot {R}_{B}^{2} = 64 \pi \therefore {R}_{B}^{2} = 64$

$\therefore {R}_{B} = \sqrt{64} = 8.0$ Centre :$\left({x}_{2} = 7 , {y}_{2} = 2\right)$

Distance between centres d= sqrt((x_1-x_2)^2+(y_1-y_2)^2

$d = \sqrt{{\left(2 - 7\right)}^{2} + {\left(3 - 2\right)}^{2}} = \sqrt{26} \approx 5.1$

${R}_{B} + {R}_{A} = 8 + 4.24 \approx 12.24$ and

${R}_{B} - {R}_{A} = 8 - 4.24 \approx 3.76 \therefore 3.76 < 5.1 < 12.24$

If $d$ lies in between $\left({R}_{B} - {R}_{A}\right) \mathmr{and} \left({R}_{B} + {R}_{A}\right)$ then circles

will intersect or overlap and have one common chord. [Ans]