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

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SUMMARY

The discussion focuses on determining the intersection points of the line defined by the equation y = -5 and the circle represented by (x - 3)² + (y + 2)² = 25. By substituting y = -5 into the circle's equation, the resulting quadratic equation (x - 3)² = 16 reveals two solutions for x: 7 and -1. Consequently, the line intersects the circle at two distinct points: (7, -5) and (-1, -5).

PREREQUISITES
  • Understanding of algebraic equations and their manipulation
  • Knowledge of the standard form of a circle's equation
  • Familiarity with quadratic equations and their solutions
  • Ability to graphically interpret intersections of lines and circles
NEXT STEPS
  • Study the derivation of the standard form of a circle's equation
  • Learn how to solve quadratic equations using the quadratic formula
  • Explore graphical methods for finding intersections of curves
  • Investigate the implications of multiple intersection points in geometry
USEFUL FOR

Students, educators, and anyone interested in algebra, particularly those studying geometry and the relationships between linear and circular equations.

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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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