gen-define-conic-sections-and-give-one-short-worked-example-9732

Define conic sections and give one short worked example.

# Conic Sections **Conic sections** are curves formed by the intersection of a plane with a double cone. There are four types: 1. **Circle** - plane perpendicular to the cone's axis 2. **Ellipse** - plane at an angle, cutting through one cone 3. **Parabola** - plane parallel to the slant of the cone 4. **Hyperbola** - plane cutting through both cones Each can be defined algebraically by second-degree equations in x and y. ## Worked Example: Circle **Problem:** Find the equation of a circle with center (2, -3) and radius 5. **Solution:** The standard form of a circle is: **(x - h)² + (y - k)² = r²** where (h, k) is the center and r is the radius. Substituting h = 2, k = -3, and r = 5: (x - 2)² + (y - (-3))² = 5² **(x - 2)² + (y + 3)² = 25** This is the equation of the circle. To verify, the point (7, -3) should lie on the circle since it's 5 units from the center: - (7 - 2)² + (-3 + 3)² = 25 + 0 = 25 ✓