# Thread: Circle Cordinate-Geomerty question, intersecting circles?

1. ## Circle Cordinate-Geomerty question, intersecting circles?

Any chance I could get a little help with the second part of this question, I've included my working for the first part in case I've messed it up.
Many thanks to anyone who can help me with this in advance!

2. Originally Posted by LHS
Any chance I could get a little help with the second part of this question, I've included my working for the first part in case I've messed it up.
Many thanks to anyone who can help me with this in advance!

You have two centers: $C_1(3, -4)$ and $C_2(15, 12)$ and the radii of the two circles are equal: $r_1 = r_2 = 10$

Since $|\overline{C_1C_2}| = 20 = 2r$ the circles are tangent to each other. The tangent point is the midpoint of $|\overline{C_1C_2}|$.

Therefore $T(9, 4)$

3. Thank you! That is very helpful. I thought it may have been something like that, as it said "the point where they touch"
For future reference is there anyway you can work out where 2 circles intersect like you would would say a circle and a line, but substitution?

4. Originally Posted by LHS
Thank you! That is very helpful. I thought it may have been something like that, as it said "the point where they touch"
For future reference is there anyway you can work out where 2 circles intersect like you would would say a circle and a line, but substitution?
Let $c_1: (x-x_C)^2+(y-y_C)^2 = r^2$ denote the first circle and $c_2: (x-x_M)^2+(y-y_M)^2=R^2$ the second circle.

Expand the brackets and subtract the first equation from the second columnwise:

$x^2-2x_Cx+x_C^2+y^2-2y_Cy +y_C^2 = r^2$

$x^2-2x_Mx+x_M^2+y^2-2y_My +y_M^2 = R^2$
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$(2x_C-2x_M)x +(2y_C-2y_M)y=R^2-r^2$

The last line is the equation of the line passing through the points of intersection. That means the problem is reduced to the calculation of points of intersection between a circle and a line.

5. Ah I see, thank you, I can see that being helpful in future questions