Two circles have the following equations (x+5)2+(y+6)2=9 and (x+2)2+(y1)2=81. 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?

1 Answer

The circles intersect but neither one of them contains the other.
Greatest possible distance df=19.615773105864 units

Explanation:

The given equations of the circle are
(x+5)2+(y+6)2=9 first circle
(x+2)2+(y1)2=81 second circle

We start with the equation passing thru the centers of the circle

C1(x1,y1)=(5,6) and C2(x2,y2)=(2,1) are the centers.

Using two-point form

yy1=(y2y1x2x1)(xx1)

y6=(1625)(x5)

y+6=(1+62+5)(x+5)

y+6=(73)(x+5)

After simplification

3y+18=7x+35

7x3y=17 equation of the line passing thru the centers and at the two points farthest to each other.

Solve for the points using first circle and the line
(x+5)2+(y+6)2=9 first circle
7x3y=17 the line

One point at A(xa,ya)=(6.1817578957376,8.7574350900543)
Another at B(xb,yb)=(3.8182421042626,3.2425649099459)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Solve for the points using second circle and the line
(x+2)2+(y1)2=81 second circle
7x3y=17 the line

One point at C(xc,yc)=(1.5452736872127,9.2723052701629)
Another at D(xd,yd)=(5.5452736872127,7.2723052701625)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
To compute for the farthest distance df we will use point A and C

df=(xaxc)2+(yayc)2

df=(6.18175789573761.5452736872127)2+(8.75743509005439.2723052701629)2

df=19.615773105864 unit)s

Kindly see the graph
Desmos.comDesmos.com

God bless .... I hope the explanation is useful.