Two circles have the following equations: `(x -6 )^2+(y -4 )^2= 64` and `(x +6 )^2+(y -9 )^2= 49`. 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?

The distance between the centers of the circles is

`d_(C)=sqrt((6-(-6))^2+(4-9)^2)=sqrt(144+25)=sqrt169=13`

The sum of the radii is

`r_1+r_2=8+7=15`

As

The distance between the centers is `<` than the sum of the radii

`d_(C)<(r_1+r_2)`,

The circles overlap.

The greatest distance is

`d_(max)=13+8+7=28u`

graph{((x-6)^2+(y-4)^2-64)((x+6)^2+(y-9)^2-49)=0 [-28.86, 28.85, -10.16, 18.7]}

`"what we have to do here is compare the distance (d )"`
`"between the centres of the circles to the "`
`color(blue)"sum/difference of the radii"`

`• " if difference of the radii > d then one circle is"`
`"contained within the other"`

`• " if sum of radii > d then circles overlap"`

`• " if sum of radii < d then no overlap"`

`• " the equation of a circle in standard form is"`

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

`(x-6)^2+(y-4)^2=64to(6,4),r=8`

`(x+6)^2+(y-9)^2to(-6,9),r=7`

`"to calculate d use the "color(blue)"distance formula"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)|)))`

`(x_1,y_1)=(6,4),(x_2,y_2)=(-6,9)`

`rArrd=sqrt((-6-6)^2+(9-4)^2)=sqrt(144+25)=13`

`• " difference of radii "=8-7=1`

`"since difference of radii < d then circle not "`
`"contained in the other"`

`"sum of radii "=8+7=15`

`"since sum of radii > d then circles overlap"`
graph{(y^2-8y+x^2-12x-12)(y^2-18y+x^2+12x+68)=0 [-40, 40, -20, 20]}