Solve System of Equations: X^2+y^2 and (x+3)^2+(y-3)^2=9

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SUMMARY

The discussion focuses on solving the system of equations represented by the circles defined by X^2 + y^2 = 9 and (x + 3)^2 + (y - 3)^2 = 9. The solution can be approached both algebraically and geometrically. The key steps involve setting the equations equal to each other, simplifying, and analyzing the distance between the centers of the circles to determine the number of intersection points. The geometric method emphasizes the importance of the midpoint between the circle centers and the perpendicular line to find the solutions.

PREREQUISITES
  • Understanding of systems of equations
  • Knowledge of circle equations in the Cartesian plane
  • Familiarity with geometric concepts such as midpoints and slopes
  • Basic algebraic manipulation skills
NEXT STEPS
  • Learn how to set up and solve systems of equations algebraically
  • Study the geometric interpretation of circle intersections
  • Explore the concept of midpoints and slopes in coordinate geometry
  • Investigate algorithms for finding intersections of geometric shapes
USEFUL FOR

Students and educators in mathematics, particularly those focusing on algebra and geometry, as well as anyone interested in solving systems of equations involving circles.

ironluis
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I need your help for solved this. :confused:

X^2+y^2=9
(x+3)^2+(y-3)^2=9

Please help me.
 
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Re: I need help!

Hi ironluis,

Welcome to MHB! (Wave)

Is this a question about systems of equations? Are you supposed to look at the two equations and solve for $x$ and $y$? This can be done algebraically however you posted the question in our geometry forum. Are you supposed to solve it geometrically?
 
Im sorry :confused:
 
Notice both equations are equal to 9, so they are equal to each other. Set them equal to each other, expand, simplify...
 
A more geometric approach would be to consider the point midway between the center of the circles. We use the mid-point in this case because the radii of the circles is the same. If the distance of this midpoint to the radii is greater than the radii, then there is no solution. If this distance is equal to the radii, there is one solution, and if it is less than the radii, and greater than zero, then there are two solutions. If the distance is zero, then the circles are concurrent and there are an infinite number of solutions.

Next, compute the slope of the line segment connecting the center of the circles, and observe that the solutions will lie along the line perpendicular to this segment, and passing through the mid-point of the centers.

This line will give you the result suggested by Prove It's much more straightforward algebraic approach.

Then you want to find the points on this line which satisfy either of the equations describing the circles.
 
Hi,
I think the best geometric solution to the problem of intersection of two circles is given at Circle, Cylinder, Sphere
I have used this algorithm with good success.
 

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