Question

Two circles have the following equations `(x +2 )^2+(y -5 )^2= 64` and `(x +4 )^2+(y -3 )^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?

Answer 1

circles intersect

Explanation:

If one circle is contained within the other then the following condition has to apply.

• d < R - r

d is the distance between the centres , R is the larger radius and r, the smaller.

The equation of a circle in standard form is.

`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 centre and r, the radius.

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

`(x+4)^2+(y-3)^2=49" has centre" (-4,3)" and "r=7`

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 points"`

The 2 points here are (-2 ,5) and (-4 ,3)

`d=sqrt((-4+2)^2+(3-5)^2)=sqrt8≈2.828`

Now R - r = 8 - 7 = 1

here d > R - r , hence circle not contained in other.

The other possibilities are as follows.

• If sum of radii > d , then circles intersect

• If sum of radii < d , then no intersection

sum of radii = 8 + 7 = 15 > d , hence circles intersect
graph{(y^2-10y+x^2+4x-35)(y^2-6y+x^2+8x-24)=0 [-40, 40, -20, 20]}