+50 Hello, here's another problem of the week!
What is the shortest distance, in units, between the circles \((x-9)^2 + (y-5)^2 = 6.25\) and \((x+6)^2 + (y+3)^2 = 49\)? Express your answer as a decimal to the nearest tenth.
+4609 This is a bit tough, but here's my take!
The shortest distance between two circles, is: \(C_1C_2-r_1-r_2\).
We have \(C_1\) as (9,5) as we take out the negatives from (-9,-5), and the radius is 2.5.
And, we have \(C_2\) as (-6,-3) as we add the negatives from (6,3), and the radius is 7.
Now, we use the distance formula! \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) Plugging in the value we get,
\(\sqrt{(-6-9)^2+(-3-5)^2}=\sqrt{225+64}=\sqrt{289}=17\). After this, we subtract both radii, to attain:
\(17-2.5-7=17-9.5=\boxed{7.5}\).
