Math CentralQuandaries & Queries


A Question from Sandra, a student:

If the center of circle A = (-1,-3) and the radius of it is √20, the center of circle B = (5,9) and the radius of it is √80. Find the point of contact of the circles A and B.

Hi Sandra,

First you need to find the equations of each of the circles then you need to solve the system of equations. I'll use a similar example where the equations are already found:

(x - 2)2 + (y - 5)2 = 25

(x - 6)2 + (y - 13)2 = 65

First expand the binomials and collect like terms to one side of the equation:

x2-4x+4 + y2 -10y+25=25 ⇒x2-4x+y2 -10y+4=0 (Eq. 1)

x2-12x+36+y2-26y+169=65 ⇒ x2-12x+y2-26y+140=0 (Eq. 2)

Since both equations now equal 0, we can set them equal to each other:

x2-4x+y2 -10y+4=x2-12x+y2-26y+140

Next collect like terms


then solve for a variable


x=17-2y (Eq. 3)

Substitute into one of the expanded equations (I choose eq. 1)

x2-4x+y2 -10y+4=0 ⇒ (17-2y)2-4(17-2y)+y2 -10y+4=0 ⇒ 289-68y+4y2-68+8y+y2-10y+4=0

and then collect like terms


We can factor this binomial and then find the roots


5(y-9)(y-5)=0 ⇒ y=9 and y=5

these are the y-values of our intersection points. to find the corresponding x-values, substitute into Eq.3



So our intersection points are (-1,9) & (7,5)

Hope this helps,


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