MHB Find the points of intersection of a line and a circle

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How do I algebraically prove how many times the line y=-5 intersects the circle (x-3)^2 + (y+2)^2 =25?
 
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What do you think you should do with y?
 
you can put y = -5 to solve for x

we get $(x-3)^2 + (-5+2)^2 = (x-3)^3+ 9 = 25$ or $(x-3)^2 = 16$
now you can solve to get x = 3 + 4 = 7 or 3-4 = - 1 so it intersects at 2 points (7,-5) and (-1,-5)
 

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