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!
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!
Your calculations are OK.
You have two centers: $\displaystyle C_1(3, -4)$ and $\displaystyle C_2(15, 12)$ and the radii of the two circles are equal: $\displaystyle r_1 = r_2 = 10$
Since $\displaystyle |\overline{C_1C_2}| = 20 = 2r$ the circles are tangent to each other. The tangent point is the midpoint of $\displaystyle |\overline{C_1C_2}| $.
Therefore $\displaystyle T(9, 4)$
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 $\displaystyle c_1: (x-x_C)^2+(y-y_C)^2 = r^2$ denote the first circle and $\displaystyle c_2: (x-x_M)^2+(y-y_M)^2=R^2$ the second circle.
Expand the brackets and subtract the first equation from the second columnwise:
$\displaystyle x^2-2x_Cx+x_C^2+y^2-2y_Cy +y_C^2 = r^2$
$\displaystyle x^2-2x_Mx+x_M^2+y^2-2y_My +y_M^2 = R^2$
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$\displaystyle (2x_C-2x_M)x +(2y_C-2y_M)y=R^2-r^2$
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.