Position of a Point with Respect to a Circle

Position of a Point with Respect to a Circle:

1. Circle Equation:

  • The equation of a circle in the Cartesian plane is x2+y2=r2 where (x,y) are coordinates and r is the radius.

2. Point-Inside, Outside, or On the Circle:

  • Inside the Circle: A point (x,y) is inside the circle if x2+y2<r2.
  • Outside the Circle: A point (x,y) is outside the circle if x2+y2>r2.
  • On the Circle: A point (x,y) lies on the circle if x2+y2=r2.

3. Distance from the Circle's Center:

  • Distance Formula: For a point (x1,y1) and the circle's center (h,k), the distance is calculated as (x1h)2+(y1k)2.
  • Comparison to Radius: If the calculated distance is:
    • Less than the radius r, the point is inside the circle.
    • Greater than r, the point is outside the circle.
    • Equal to r, the point lies on the circle.

4. Using Inequality to Determine Position:

  • Inequality Test: For a point (x1,y1), compare x12+y12 with r2.
  • Inside or Outside Check: If x12+y12<r2, the point is inside; if x12+y12>r2, it's outside.
  • On the Circle Check: If x12+y12=r2, the point lies on the circle.

5. Geometric Interpretation:

  • Visualizing the Circle: Plotting the circle on the Cartesian plane helps understand the point's position relative to the circle.
  • Understanding Quadrants: Analyzing points in relation to the circle's center and quadrants helps determine their positions.

6. Applications:

  • Geometry and Trigonometry: Used in understanding geometric relationships and applying trigonometric concepts.
  • Real-world Scenarios: Applicable in navigation, mapping, and determining object positions.