Overview

This topic covers conic sections and their equations, as well as key geometric relationships in the coordinate plane. You'll learn to analyze lines, circles, parabolas, ellipses, and hyperbolas using algebraic tools.

Key Concepts and Equations

• Distance Formula: d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
• Midpoint Formula: M = ((x_1 + x_2)/2, (y_1 + y_2)/2)
• Slope: m = (y_2 - y_1)/(x_2 - x_1)
• Equation of a Line:
  • Slope-intercept form: y = mx + b
  • Point-slope form: y - y_1 = m(x - x_1)
• Circle: (x - h)^2 + (y - k)^2 = r^2
• Parabola (vertical): y = a(x - h)^2 + k
• Ellipse: \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
• Hyperbola: \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
• Foci and Axes: Understand location and roles in ellipses and hyperbolas

Step-by-Step Example

Problem: Write the equation of a circle with center at (3, -2) and radius 5.

Step 1: Use the circle equation:
(x - h)^2 + (y - k)^2 = r^2

Step 2: Substitute h = 3, k = -2, and r = 5:
(x - 3)^2 + (y + 2)^2 = 25

Answer: (x - 3)^2 + (y + 2)^2 = 25

Quick Tip