Question

Two circles have the following equations `(x +5 )^2+(y -2 )^2= 36` and `(x -1 )^2+(y -1 )^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

circles overlap. 21.08

Explanation:

The standard form of the`color(blue)" 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 the standard equation.

`(x+5)^2+(y-2)^2=36" has centre (-5,2) and r = 6"`

`(x-1)^2+(y-1)^2=81" has centre (1,1) and r = 9"`

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)=(-5,2)" and " (x_2,y_2)=(1,1)`

`rArr d=sqrt((1+5)^2+(1-2)^2)=sqrt(36+1)=sqrt37 ≈ 6.08`

Now sum of radii = 6 + 9 = 15

Since sum of radii > d , then circles overlap.

greatest distance = d + sum of radii = 6.08 + 15 = 21.08
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