Intercepts Made by a Circle on Axes

Intercepts Made by a Circle on Axes:

1. Definition:

  • X-Intercepts: The points where a circle intersects the x-axis.
  • Y-Intercepts: The points where a circle intersects the y-axis.

2. X-Intercepts:

  • Definition: The x-intercepts are points where the circle touches or crosses the x-axis.
  • Coordinate Form: X-intercepts occur at points (a,0) and (a,0) where a is the distance from the circle's center to the x-intercept.
  • Equation Relation: For a circle with equation x2+y2=r2, the x-intercepts are at (±r,0).

3. Y-Intercepts:

  • Definition: The y-intercepts are points where the circle touches or crosses the y-axis.
  • Coordinate Form: Y-intercepts occur at points (0,b) and (0,b) where b is the distance from the circle's center to the y-intercept.
  • Equation Relation: For a circle with equation x2+y2=r2, the y-intercepts are at (0,±r).

4. Calculation from Circle Equation:

  • Standard Form: x2+y2=r2 represents a circle centered at the origin (0,0) with radius r.
  • Using Radius: X-intercepts are at (±r,0) and y-intercepts at (0,±r).

5. Circle Equation with Center at (h,k):

  • Shifted Circle Equation: (xh)2+(yk)2=r2 represents a circle with center (h,k) and radius r.
  • Intercepts Calculation: For this equation, the x-intercepts are at (h±r,k) and y-intercepts at (h,k±r).

6. Significance and Applications:

  • Geometry and Graphing: Understanding intercepts aids in plotting circles on the Cartesian plane.
  • Problem Solving: Intercepts provide valuable information about the circle's position and size.

7. Example:

  • Circle Equation: x2+y2=25 represents a circle with radius r=5.
  • Intercepts Calculation:
    • X-Intercepts: At (±5,0).
    • Y-Intercepts: At (0,±5).