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)
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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