Question

Two circles have the following equations `(x -1 )^2+(y -2 )^2= 64` and `(x +7 )^2+(y +2 )^2= 9`. 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

circles overlap.

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 difference of radii > d , then 1 circle contained in other

To find the centres and radii of the circles, using equation.

`color(orange)"Reminder " 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-1)^2+(y-2)^2=64" has centre (1,2) and r = 8"`

`(x+7)^2+(y+2)^2=9" has centre (-7,-2) and r = 3"`

To find 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"`

The 2 points here are (1 ,2) and (-7 ,-2)

let `(x_1,y_1)=(1,2)" and " (x_2,y_2)=(-7,-2)`

`d=sqrt((-7-1)^2+(-2-2)^2)=sqrt80≈8.944`

sum of radii = 8 + 3 = 11

Since sum of radii > d , then circles overlap
graph{(y^2-4y+x^2-2x-59)(y^2+4y+x^2+14x+44)=0 [-40, 40, -20, 20]}