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# Coordinates

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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?

Apr 25, 2022

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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!!!

Apr 25, 2022
edited by BuilderBoi  Apr 25, 2022