The Coordinate Geometry of Circles: x^2 + y^2 = r^2

The Coordinate Geometry of Circles: x^2 + y^2 = r^2

Understanding the Equation of a Circle

  • The general equation for a circle in the 2-dimensional coordinate system is x² + y² = r², where (0, 0) is the center of the circle and r is the radius.
  • This formula is derived from Pythagoras’ Theorem and embodies the definition of a circle as a set of points equidistant from a fixed point (the center).

Center at the Origin:

  • When the center of the circle (h, k) is at the origin of the coordinate system (0,0), the formula simplifies to x² + y² = r².
  • Here, r represents the radius of the circle.

Shifting the Center:

  • The equation changes when the center of the circle moves to a point other than the origin.
  • The equation of a circle centered at the point (h, k) and with the radius r is (x - h)² + (y - k)² = r².
    • Here, h and k are the x and y coordinates of the center of the circle respectively.

Finding the Radius and Center:

  • To find the center and the radius of a circle from its equation, you have to rewrite the equation in the standard form.
    • The coefficients of x² and y² should be the same and equal to 1.
    • The signs before the x-term and the y-term should be negative.

Graphing a Circle:

  • To graph a circle, begin by plotting the center point (h, k).
  • From this point, use the radius to mark out the boundary of your circle.
  • Always sketch lightly first, as you may need to adjust your circle.

Intersections of Lines and Circles:

  • A line and a circle will intersect at a point where the equation of the line satisfies the equation of the circle.
  • By substituting the y-value of the line into the circle equation, you can solve for the x-values of the intersection points, and then find the corresponding y-values.