Class 11 Notes - Circles

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Circles – Class 11

In this chapter, we explore the advanced properties of circles, their equations, and their applications in coordinate geometry. Understanding circles is fundamental for higher mathematics, engineering, and real-world problem solving.

1. Definition and Equation of a Circle

  • Circle: The set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).
  • Standard Equation: For a circle with center at (h, k) and radius r:
    (x - h)2 + (y - k)2 = r2
  • Special Case: Center at origin (0, 0): x2 + y2 = r2

2. General Equation of a Circle

  • General Form: x2 + y2 + 2gx + 2fy + c = 0
    Center: (-g, -f), Radius: √(g2 + f2 - c)

3. Position of a Point Relative to a Circle

  • Substitute the point (x1, y1) into the circle's equation:
    • If LHS < r2: Point is inside the circle
    • If LHS = r2: Point is on the circle
    • If LHS > r2: Point is outside the circle

4. Tangent to a Circle

  • Definition: A line that touches the circle at exactly one point.
  • Equation of Tangent: At point (x1, y1) on the circle:
    x1x + y1y = r2 (for center at origin)
  • Length of Tangent from a Point (x1, y1) to the Circle:
    √[(x1 - h)2 + (y1 - k)2 - r2]

5. Chord of a Circle

  • Chord: A line segment joining any two points on the circle.
  • Equation of Chord with Midpoint (x1, y1):
    T = S1

6. Intersection of a Line and a Circle

  • Substitute the line equation into the circle's equation and solve the resulting quadratic to find intersection points.
  • Number of Intersection Points:
    • 2 points: Line is a secant
    • 1 point: Line is a tangent
    • 0 points: Line does not intersect the circle

7. Family of Circles

  • Circles passing through the intersection of two given circles:
    S1 + λS2 = 0

8. Radical Axis and Radical Center

  • Radical Axis: The locus of points having equal power with respect to two circles.
  • Radical Center: The point of intersection of the radical axes of three circles.

9. Practice Problems

  1. Find the equation of a circle with center (2, -3) and radius 5.
  2. Determine whether the point (4, 1) lies inside, on, or outside the circle x2 + y2 = 20.
  3. Find the length of the tangent from the point (7, 4) to the circle x2 + y2 = 25.
  4. Write the equation of the tangent to the circle x2 + y2 = 16 at the point (2, 2√3).
  5. Find the equation of the chord of the circle x2 + y2 = 9 whose midpoint is (1, 2).

10. Summary

  • Standard and general equations of a circle
  • Position of a point relative to a circle
  • Tangents and chords
  • Intersection of lines and circles
  • Radical axis and radical center
  • Applications in coordinate geometry