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Conic Sections Master Class

From The Cone to The Equations

The Circle Horizontal Cut

Slice the cone horizontally (perpendicular to the axis). The distance from the center is constant.

circle

The Ellipse Angled Cut

Tilt the slice. It cuts through both sides but at an angle, stretching the circle into an oval.

Ellipse

Locus Definition: Pins & Strings

Interact: Drag the colored point P along the curve. The strings will update.

Circle (One Pin)

Fixed distance ($r$) from Center $(h,k)$.

Ellipse (Two Pins)

Sum of distances to two Foci is constant ($d_1 + d_2 = 2a$).

Algebraic Anatomy

Circle
$$(x-h)^2 + (y-k)^2 = r^2$$
  • $(h,k)$: Center
  • $r$: Radius
  • Area: $\pi r^2$
Ellipse
$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$

⚠️ RHS must be 1

  • $a$: Semi-major axis (Horizontal)
  • $b$: Semi-minor axis (Vertical)
  • Foci ($c$): $c^2 = a^2 - b^2$ (Use largest - smallest)
  • Area: $\pi a b$

Skill 1: Completing the Square (Ellipse)

Convert: $9x^2 + 4y^2 - 18x = 27$

  1. Group and factor: $9(x^2 - 2x) + \dots$
  2. Complete the square.
  3. Divide to make RHS = 1.

Skill 2: Tangents (Implicit Diff)

Rule: $\frac{d}{dx}(y^2) = 2y \cdot y'$.

Find tangent to $\frac{x^2}{8} + \frac{y^2}{2} = 1$ at point $(2, 1)$.

🏠 Homework: Closed Loop Scenarios

1. Circle (Radar): A radar covers $x^2 + y^2 - 20x - 16y = 0$ (km). Find the radius of detection.

2. Ellipse (Tunnel): Tunnel width 12m, height 4m (center 0,0).

  • Find the tunnel height at 4m from the center.

The Parabola Parallel Cut

The plane is parallel to the slanted edge of the cone. The curve opens forever.

parabola

The Hyperbola Vertical Cut

The plane cuts vertically, slicing both the top and bottom cones (double cone).

hyperbola

Locus Definition: Distances

Interact: Drag point P. It stays on the curve.

Parabola

Distance to Focus = Distance to Line ($d_1 = d_2$).

Hyperbola

Difference of distances is constant ($|d_1 - d_2| = k$).

Algebraic Anatomy

Parabola
$$(y-k)^2 = 4a(x-h)$$
  • One squared term only.
  • $a$: Focal Length.
  • Directrix: Line at distance $a$ behind vertex.
Hyperbola
$$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

⚠️ RHS must be 1

  • Minus sign determines direction.
  • Foci ($c$): $c^2 = a^2 + b^2$ (Add them!)
  • Asymptotes: Linear guides.

Skill 1: Hyperbola Foci

For $\frac{x^2}{16} - \frac{y^2}{9} = 1$, find the Foci.

Skill 2: Tangent Equation

Find tangent to $x^2 - y^2 = 16$ at $(5, 3)$.

🏠 Homework: Open Curve Scenarios

1. Parabola (Reflector): A headlight is 20cm wide and 5cm deep. Model with vertex at (0,0).

  • Where is the Focus (bulb position)?

2. Hyperbola (Navigation): Ship position determined by signal delay: $x^2 - y^2 = 100$. Ship is at $x=20$.

  • Find the y-coordinate.