Question

Two circles have the following equations: `(x +2 )^2+(y -1 )^2= 49` and `(x +4 )^2+(y +7 )^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?

Answer 1

The circles overlap, greatest distance ≈ 24.246 units.

Explanation:

What we have to do here is compare the distance ( d) between the centres of the circles to the 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 inside the other

Comparing the given equations with the standard form of the equation of a circle.

`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=49" has centre (-2,1) and "r=7`

`(x+4)^2+(y+7)^2=81" has centre (-4,-7) and " r=9`

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)

let `(x_1,y_1)=(-2,1)" and " (x_2,y_2)=(-4,-7)`

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

Sum of radii = 7 + 9 = 16

Difference of radii = 9 - 7 =2

Since sum of radii > d , then circles overlap

greatest distance between them = d + sum of radii

`=8.246+16=24.246`
graph{(y^2-2y+x^2+4x-44)(y^2+14y+x^2+8x-16)=0 [-35.08, 35.08, -17.53, 17.55]}