Two circles have the following equations: #(x +2 )^2+(y -1 )^2= 16 # and #(x +4 )^2+(y +7 )^2= 25 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

What we have to do here is compare the distance ( d ) between the centres of the circles with the sum or difference of the radii.
There are 3 possible outcomes.

• If sum of radii > d , #color(blue)"then circles overlap"#

• If sum of radii < d , #color(blue)"then no overlap"#

• If difference of radii > d #color(red)"then 1 circle contains the other"#

The standard form of the #color(blue)"equation of a circle"# is

#color(red)(|bar(ul(color(white)(a/a)color(black)((x-a)^2+(y-b)^2=r^2)color(white)(a/a)|)))#
where (a ,b) are the coordinates of the centre and r, the radius.

#(x+2)^2+(y-1)^2=16rArr" centre"=(-2,1),r=4#

#(x+4)^2+(y+7)^2=25rArr"centre"=(-4,-7),r=5#

To calculate d, use the #color(blue)"distance formula"#

#color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))#
where # (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"#

The 2 points here are (-2 ,1) and (-4 ,-7) the centres of the circles.

#d=sqrt((-4+2)^2+(-7-1)^2)=sqrt(4+64)=sqrt68≈8.246#

sum of radii = 4 + 5 = 9

difference of radii = 5 - 4 = 1

Since sum of radii > d , then circles overlap
graph{(y^2-2y+x^2+4x-11)(y^2+14y+x^2+8x+0)=0 [-40, 40, -20, 20]}