Question

Two circles have the following equations: `(x -1 )^2+(y -2 )^2= 9` 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

circles overlap, 13.385

Explanation:

What we have to do here is calculate the distance between the centres of the circles and compare this with the sum of the radii.

Comparing the given equations with the standard form

`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) is the centre and r, the radius.

`(x-1)^2+(y-2)^2=9 " has centre (1,2) and r = 3"`

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

Calculate the distance (d) between centres using 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)|)))`

`rArrd=sqrt((-4-1)^2+(-2-2)^2)=sqrt29 ≈ 5.385`

sum of radii = 3 + 5 = 8

Since sum of radii > d , then circles overlap meaning one circle does not contain the other.

and greatest possible distance
= d + sum of radii
= 5.385 + 8 = 13.385
graph{(y^2-4y+x^2-2x-4)(y^2+4y+x^2+8x-5)=0 [-20, 20, -10, 10]}