What is the product of all the coordinates of all the points of intersection of the two circles defined by x^2-2x+y^2-10y+20=0 and x^2-8x+y^2-10y+35=0?
+2668 This is going to be a lot of work...
We first start by setting these equations equal to each other, so the \(x^2\) and \(y^2\) cancel out:\(x^2-2x+y^2-10y+20=-10x-20y-15\)
Setting the left-hand side equal to 0, we get: \(-10x-20y-15=0\)
We want another equation without any exponents to solve for x.
We can get this other equation by subtracting the original equations. This yields: \(16x+20y=0\)
Solving for x, we find that \(x = 2.5\). This means both points of the intersection occur when \(x = 2.5\)
Subbing x back into the first circle, we get the quadratic: \(y^2-10y+21.25=0\)
Solving, we find \(y = 5 \pm {\sqrt{15} \over 2}\)
Thus, the 2 co-ordinates of intersection are: \((2.5, {5 + {\sqrt{15} \over 2}})\) and \((2.5, {5 - {\sqrt{15} \over 2}})\)
To find the answer, you need to multiply all 4 values together, feel free to ask if you need any help!!!
