Intersection of a Line and a Circle

Intersection of a Line and a Circle:

1. Equation of a Line:

  • A line is represented in the Cartesian plane by the equation y=mx+c (slope-intercept form) or Ax+By+C=0 (general form).

2. Equation of a Circle:

  • The equation of a circle is x2+y2=r2 for a circle centered at the origin or (xh)2+(yk)2=r2 for a circle centered at (h,k).

3. Intersection Conditions:

  • Geometric Concept: The intersection of a line and a circle occurs when the line passes through, is tangent to, or doesn’t touch the circle at all.
  • Mathematical Condition: For intersection, the coordinates satisfying both the line equation and the circle equation simultaneously represent the intersection points.

4. Solving for Intersection Points:

  • Example Equations:

    • Circle Equation: (x2)2+(y3)2=52for a circle centered at (2,3) with radius 5.
    • Line Equation: y=2x+1 (slope-intercept form).
  • Substitution Method: Substitute y=2x+1 into the circle equation to form a quadratic equation in terms of x.

    • Substitute y in the circle equation: (x2)2+(2x+13)2=52.
    • Solve for x to find possible x-coordinates.
  • Using Quadratic Formula: After obtaining the x-coordinates, substitute them back into the line equation to find corresponding y-coordinates.

    • If x is 2, then y=2×2+1=5 (one intersection point).

5. Geometric Interpretation:

  • Visual Analysis: Plotting the line and circle on a graph helps visualize their intersection points or lack thereof.
  • Understanding Tangency: A line tangent to a circle intersects it at a single point, whereas a line passing through the circle has two intersection points.

6. Special Cases:

  • No Intersection: If the discriminant of the quadratic equation (b24ac) is negative, there is no real intersection (line doesn’t intersect the circle).
  • Tangency: A line tangent to the circle has a single point of intersection with the circle.