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

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To find the points of intersection between the line y = -5 and the circle defined by (x-3)² + (y+2)² = 25, substitute y = -5 into the circle's equation. This leads to the equation (x-3)² + (-5+2)² = 25, simplifying to (x-3)² + 9 = 25. Further simplification gives (x-3)² = 16, allowing for the solution of x values. The resulting x values are x = 7 and x = -1, indicating that the line intersects the circle at two points: (7, -5) and (-1, -5). Thus, the line intersects the circle at two distinct points.
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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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