Circles A and B have the following equations #(x -4 )^2+(y -8 )^2= 25 # and #(x +4 )^2+(y -1 )^2= 49 #. What is the greatest possible distance between a point on circle A and another point on circle B?

Question

Circles A and B have the following equations #(x   -4  )^2+(y   -8   )^2=  25  # and #(x    +4   )^2+(y   -1   )^2=  49  #.  What is the greatest possible distance between a point on circle A and another point on circle B?

Mason Adams · Accepted Answer

#5+sqrt113+7=22.63# distance between circle A center and circle B center = #sqrt(8^2+7^2)=sqrt113# So the greatest possiible distance between a point on circle A and a point on circle B should be the straight line passing through the centers points of these two circles #= 5+sqrt113+7 = 22.63#

Caroline Aucoin · Answer

undefined To find the greatest possible distance between a point on circle A and another point on circle B, we need to find the distance between their centers and then subtract the sum of their radii.

The center of circle A is (4, 8) and the center of circle B is (-4, 1). Using the distance formula, the distance between their centers is:

$$ \sqrt{(4 - (-4))^2 + (8 - 1)^2} = \sqrt{8^2 + 7^2} = \sqrt{64 + 49} = \sqrt{113} $$

The radius of circle A is $ \sqrt{25} = 5 $ and the radius of circle B is $ \sqrt{49} = 7 $.

So, the greatest possible distance between a point on circle A and another point on circle B is:

$$ \sqrt{113} - (5 + 7) = \sqrt{113} - 12 $$