Question

Circle A has a center at `(1 ,1 )` and a radius of `1`. Circle B has a center at `(2 ,-3 )` and a radius of `5`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

points of intersection: `((27+4sqrt(19))/34,(62+sqrt(19))/34) and ((27-4sqrt(19))/34,(62-sqrt(19))/34)` or approximately `(1.307,1.952)` and `(0.281,1.695)`.

Explanation:

Recall that the general equation of a circle is:

`(x-h)^2+(y-k)^2=r^2`

where:
`x=`any x-coordinate of the circle
`h=`x-coordinate of the circle's center
`y=`any y-coordinate of the circle
`k=`y-coordinate of the circle's center
`r=`radius

`1`. Determine the equation of each circle:

Circle A
`(x-1)^2+(y-1)^2=1^2`

Circle B
`(x-2)^2+(y+3)^2=5^2`

`2`. Simplify the equations so that they both equal `0`.

Circle A
`(x-1)^2+(y-1)^2=1^2`

`(x-1)(x-1)+(y-1)(y-1)=1`

`(x^2-2x+1)+(y^2-2y+1)=1`

Equation `1`: `color(orange)(x^2+y^2-2x-2y+1=0)`

Circle B
`(x-2)^2+(y+3)^2=5^2`

`(x-2)(x-2)+(y+3)(y+3)=25`

`(x^2-4x+4)+(y^2+6y+9)=25`

Equation `2`: `color(blue)(x^2+y^2-4x+6y-12=0)`

`3`. Since the equations both equal `0`, they must both be equal to each other.

`color(orange)(x^2+y^2-2x-2y+1)=color(blue)(x^2+y^2-4x+6y-12)`

`4`. Simplify.

`color(red)cancelcolor(orange)(x^2)color(red)cancelcolor(orange)(+y^2)color(orange)(-2x-2y+1)=color(red)cancelcolor(blue)(x^2)color(red)cancelcolor(blue)(+y^2)color(blue)(-4x+6y-12)`

`color(orange)(-2x-2y+1)=color(blue)(-4x+6y-12)`

`5`. Solve for `y`.

`color(orange)(-2y)` `color(blue)(-6y)=color(blue)(-4x)` `color(orange)(+2x)` `color(blue)(-12)` `color(orange)(-1)`

`-8y=-2x-13`

`(-8y)/(-8)=(-2x)/(-8)-13/(-8)`

`color(purple)(y=1/4x+13/8)`

`6`. Substitute `color(purple)(y=1/4x+13/8)` into either equation `1` or `2`. In this case, we'll use equation `1`.

`color(orange)(x^2+color(purple)y^2-2x-2color(purple)y+1=0)`

`x^2+color(purple)((1/4x+13/8))^2-2x-2color(purple)((1/4x+13/8))+1=0`

`7`. Simplify.

`x^2+color(purple)((1/4x+13/8)(1/4x+13/8))-2x-2color(purple)((1/4x+13/8))+1=0`

`x^2+(1/16x^2+13/16x+169/64)-2x-color(red)cancelcolor(black)2^1color(purple)((1/color(red)cancelcolor(purple)4^2x+13/color(red)cancelcolor(purple)8^4))+1=0`

`17/16x^2+13/16x+169/64-2x-1/2x-13/4+1=0`

`17/16x^2-27/16x+25/64=0`

`8`. Use the quadratic formula to factor the equation. This will tell you the x-coordinates of where the two circles intersect.

`color(turquoise)(a=17/16)color(white)(XXX)color(gold)(b=-27/16)color(white)(XXX)color(brown)(c=25/64)`

`x=(-b+-sqrt(b^2-4ac))/(2a)`

`x=(-(color(gold)(-27/16))+-sqrt((color(gold)(-27/16))^2-4(color(turquoise)(17/16))(color(brown)(25/64))))/(2(color(turquoise)(17/16)))`

`x=(27/16+-sqrt(729/256-425/256))/(17/8)`

`x=(27/16+-sqrt(304/256))/(17/8)`

`x=(27/16+-sqrt(19)/4)/(17/8)`

`x=(27/16+-(4sqrt(19))/16)xx8/17`

`x=((27+-4sqrt(19))/color(red)cancelcolor(black)16^2)xxcolor(red)cancelcolor(black)8^1/17`

`color(green)(|bar(ul(color(white)(a/a)x=(27+-4sqrt(19))/34color(white)(a/a)|)))`

`9`. Now that you have the x-coordinates of the points of intersection, substitute each x-coordinate into `color(purple)(y=1/4x+13/8)` to find the corresponding y-coordinates.

Intersection 1
`color(purple)(y=1/4x+13/8)`

`y=1/4((27+4sqrt(19))/34)+13/8`

`y=(27+4sqrt(19))/136+13/8`

`y=(27+4sqrt(19))/136+(17(13))/136`

`y=(27+4sqrt(19))/136+221/136`

`y=(248+4sqrt(19))/136`

`y=(4(62+sqrt(19)))/(4(34))`

`y=(color(red)cancelcolor(black)4(62+sqrt(19)))/(color(red)cancelcolor(black)4(34))`

`color(green)(|bar(ul(color(white)(a/a)y=(62+sqrt(19))/34color(white)(a/a)|)))`

Intersection 2
`color(purple)(y=1/4x+13/8)`

`y=1/4((27-4sqrt(19))/34)+13/8`

`y=(27-4sqrt(19))/136+13/8`

`y=(27-4sqrt(19))/136+(17(13))/136`

`y=(27-4sqrt(19))/136+221/136`

`y=(248-4sqrt(19))/136`

`y=(4(62-sqrt(19)))/(4(34))`

`y=(color(red)cancelcolor(black)4(62-sqrt(19)))/(color(red)cancelcolor(black)4(34))`

`color(green)(|bar(ul(color(white)(a/a)y=(62-sqrt(19))/34color(white)(a/a)|)))`

If you graphed the two circles, it would look like:

Zoomed in, you can get a sense as to where the intersections are:

`:.`, the circles overlap at `((27+4sqrt(19))/34,(62+sqrt(19))/34) and ((27-4sqrt(19))/34,(62-sqrt(19))/34)` or at approximately `(1.307,1.952)` and `(0.281,1.695)`.