Lecture Notes: ASP Maths Curriculum - Circles

Topics Covered

• Midpoints and Perpendicular Bisectors
• Equation of a Circle
• Intersections of Straight Lines and Circles
• Tangents to Circles
• Chord Properties
• Circles with Triangles

1. Midpoints and Perpendicular Bisectors

• Example Problem: Line segment AB is the diameter of a circle centered at C. Given points A(-1, 4) and B(5, 2).
  • Find the midpoint of AB:
    • Midpoint formula: ((x_1 + x_2)/2, (y_1 + y_2)/2)
    • Midpoint = (2, 3)
  • Perpendicular Bisector:
    • Gradient of AB = ((y_2 - y_1)/(x_2 - x_1))
    • Perpendicular gradient is the negative reciprocal: if gradient of AB is -1/3, perpendicular gradient = 3.
    • Equation of line: (y - y_1 = m(x - x_1))

2. Equation of a Circle

• General Form: ((x - a)^2 + (y - b)^2 = r^2)
• Example:
  • Center at (3,1), radius squared determined using Pythagorean theorem.
  • Circle equation: (x - 3)^2 + (y - 1)^2 = 41)
• Completing the Square: Used to rearrange equations into circle form.

3. Intersections of Straight Lines and Circles

• Example Problem: Show that line does not intersect with a given circle.
  • Use simultaneous equations.
  • Solve quadratic equation:
    • Rearrange to form (ax^2 + bx + c = 0)
    • Use discriminant (b^2 - 4ac). If < 0, no real roots, hence no intersection.

4. Tangents to Circles

• Properties: Tangent is perpendicular to radius.
• Example: Find tangent at point P(1, -2) on circle centered at (4, 6).
  • Calculate the gradient of radius.
  • Negative reciprocal for tangent gradient.
  • Equation of tangent line using point-slope form.

5. Triangles in Circles

• Example Problem: Show AB is diameter of circle.
  • Use midpoint and perpendicular bisectors.
  • If AB is a diameter, triangle formed with any point on the circle is a right triangle (circle theorem).
  • Use Pythagorean theorem to confirm diameter.

Conclusion

• Covered key concepts for ASP maths curriculum related to circles.
• Reviewed midpoint, gradients, equations of circles, and properties of tangents and chords.
• Utilized circle theorems and Pythagorean theorem for problem-solving.

This concludes the lecture. Remember to review these key points to prepare for your exams and assignments. If you have any questions, please don't hesitate to reach out.