Question

Two circles have the following equations `(x +5 )^2+(y +3 )^2= 9` and `(x +4 )^2+(y -1 )^2= 16`. 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. 11.123

Explanation:

The standard form of the equation of a circle 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 coords of centre and r , the radius.

both equations here are in this form and so we can extract centres and radii by comparison with standard equation.

`(x+5)^2+(y+3)^2 = 9 " has centre = (-5,-3) and r = 3"`

`(x+4)^2+(y-1)^2 = 16" has centre = (-4,1) and r = 4"`

Calculate the distance 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)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are the coords of 2 points "`

let `(x_1,y_1)=(-4,1)" and " (x_2,y_2)=(-5,-3)`

`rArr d = sqrt((-5+4)^2+(-3-1)^2)=sqrt(1+16)`
`=sqrt17 ≈ 4.123`

now sum of radii = 3+4 = 7

Since sum of radii > distance between centres , circles overlap.

greatest distance = d + sum of radii = 4.123 + 7 = 11.123
graph{(y^2+6y+x^2+10x+25)(y^2-2y+x^2+8x+1)=0 [-20, 20, -10, 10]}