Question

Two circles have the following equations: `(x -8 )^2+(y -5 )^2= 64` and `(x +4 )^2+(y +2 )^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?

Answer 1

No overlap, greatest distance ≈ 26.89 units

Explanation:

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

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

• If diff. of radii > d , then 1 circle inside other

We require to find the centres and radii of the circles.

The standard form of the `color(blue)"equation of a circle"` 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.

`rArr(x-8)^2+(y-5)^2=64to" centre "=(8,5),r=8`

`(x+4)^2+(y+2)^2=25to"centre "=(-4,-2),r=5`

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))color(white)(2/2)|)))`
where `(x_1,y_1),(x_2,y_2)" are 2 coordinate points"`

let `(x_1,y_1)=(8,5)" and " (x_2,y_2)=(-4,-2)`

`d=sqrt((-4-8)^2+(-2-5)^2)=sqrt193≈13.89`

sum of radii = 8 + 5 = 13

and diff. of radii = 8 - 5 = 3

Since diff. of radii < d , then 1 circle NOT inside the other

Since sum of radii < d , then no overlap of circles

Greatest distance between 2 points = sum of radii + d

`=13+13.89=26.89" units (to 2 decimal places)"`
graph{(y^2-10y+x^2-16x+25)(y^2+4y+x^2+8x-5)=0 [-40, 40, -20, 20]}