Conic Sections: Ellipse & Hyperbola – Definitions, Equations & Properties

TAKEAWAYS:

• Ellipses are stretched circles with two focal points; their standard equation is $\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$, where $a>b$.
• Hyperbolas have two branches and an asymptote; their standard equations are $\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1$ (horizontal) or $\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1$ (vertical).
• Eccentricity ($e$) measures how "stretched" a conic is: $0<e<1$ for ellipses, $e>1$ for hyperbolas, and $e=1$ for parabolas.
• Applications include planetary motion (ellipses), satellite dishes (parabolas), and cooling towers (hyperbolas).
• Graphing requires identifying the center, vertices, co-vertices, foci, and asymptotes (for hyperbolas).
• NEB exam focus: Deriving equations from given conditions, identifying conics from equations, and solving word problems involving distances and foci.

1. Introduction to Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The three main types are:

• Ellipse (plane cuts the cone at an angle less than the side of the cone).
• Parabola (plane cuts the cone parallel to its side).
• Hyperbola (plane cuts the cone at an angle greater than the side of the cone).

2. Ellipse

mindmap
  root((Conic Sections))
    Ellipse
      Closed Curve
      e < 1
      Foci Inside
      Planetary Orbits
    Hyperbola
      Open Curve (2 Branches)
      e > 1
      Foci Outside
      Asymptotes
      Cooling Towers
    Parabola
      Open Curve (1 Branch)
      e = 1
      Focal Point
      Satellite Dishes

Definition

An ellipse is the set of all points $(x,y)$ such that the sum of the distances to two fixed points (foci) is constant.

Standard Equation

For an ellipse centered at $(h,k)$ with major axis parallel to the x-axis: $$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1(a>b)$$

• $a$: semi-major axis (half the length of the longest diameter).
• $b$: semi-minor axis (half the length of the shortest diameter).
• $c$: distance from the center to each focus, where $c^2=a^2-b^2$.
• Eccentricity: $e=\frac{c}{a}$ (always $0<e<1$).

For an ellipse with major axis parallel to the y-axis: $$\frac{(x-h{)}^2}{b^2}+\frac{(y-k{)}^2}{a^2}=1(a>b)$$

Key Properties

• Vertices: $(h\pm a,k)$ or $(h,k\pm a)$.
• Co-vertices: $(h,k\pm b)$ or $(h\pm b,k)$.
• Foci: $(h\pm c,k)$ or $(h,k\pm c)$.

Worked Example 1: Deriving the Equation of an Ellipse

Problem: Find the standard equation of an ellipse with foci at $(2,0)$ and $(-2,0)$, and a major axis length of 6.

Solution:

1. The foci are symmetric about the origin, so the center is $(0,0)$.
2. Distance between foci $=2c=4\Rightarrow c=2$.
3. Major axis length $=2a=6\Rightarrow a=3$.
4. Use $c^2=a^2-b^2$: $$2^2=3^2-b^2\Rightarrow 4=9-b^2\Rightarrow b^2=5$$
5. The standard equation is: $$\frac{x^2}{9}+\frac{y^2}{5}=1$$

3. Hyperbola

Definition

A hyperbola is the set of all points $(x,y)$ such that the absolute difference of the distances to two fixed points (foci) is constant.

Standard Equations

For a hyperbola centered at $(h,k)$ with a horizontal transverse axis: $$\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1$$ For a hyperbola with a vertical transverse axis: $$\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1$$

• $a$: distance from the center to each vertex.
• $b$: related to the distance from the center to the co-vertices.
• $c$: distance from the center to each focus, where $c^2=a^2+b^2$.
• Eccentricity: $e=\frac{c}{a}$ (always $e>1$).
• Asymptotes: Lines that the hyperbola approaches but never touches. For the horizontal hyperbola, the asymptotes are: $$y-k=\pm \frac{b}{a}(x-h)$$

Key Properties

• Vertices: $(h\pm a,k)$ or $(h,k\pm a)$.
• Foci: $(h\pm c,k)$ or $(h,k\pm c)$.
• Asymptotes: $y=\pm \frac{b}{a}x$ (for center at origin).

Worked Example 2: Graphing a Hyperbola

Problem: Graph the hyperbola $\frac{(x-1{)}^2}{4}-\frac{(y+2{)}^2}{9}=1$. Identify the center, vertices, foci, and asymptotes.

Solution:

1. Center: $(h,k)=(1,-2)$.
2. Vertices: $(h\pm a,k)=(1\pm 2,-2)\Rightarrow (3,-2)$ and $(-1,-2)$.
3. $a^2=4\Rightarrow a=2$, $b^2=9\Rightarrow b=3$.
4. $c^2=a^2+b^2=4+9=13\Rightarrow c=\sqrt{13}$.
5. Foci: $(1\pm \sqrt{13},-2)$.
6. Asymptotes: $y+2=\pm \frac{3}{2}(x-1)$.

4. Comparison Table: Ellipse vs. Hyperbola

5. Applications of Conic Sections

1. Ellipses:

   • Planetary orbits (e.g., Earth's orbit around the Sun).
   • Architecture (e.g., domes, arches).
   • Optics (e.g., elliptical mirrors focus light to a point).

2. Hyperbolas:

   • Cooling towers (e.g., nuclear power plants).
   • Satellite dishes (reflect signals to a focal point).
   • Navigation (e.g., GPS uses hyperbolic paths).

6. Solving Problems Involving Conic Sections

Worked Example 3: Finding the Foci of an Ellipse

Problem: Find the foci of the ellipse $4x^2+9y^2=36$.

Solution:

1. Rewrite in standard form: $$\frac{x^2}{9}+\frac{y^2}{4}=1$$
2. Here, $a^2=9\Rightarrow a=3$, $b^2=4\Rightarrow b=2$.
3. Calculate $c$: $$c^2=a^2-b^2=9-4=5\Rightarrow c=\sqrt{5}$$
4. The foci are at $(\pm \sqrt{5},0)$.

Worked Example 4: Identifying a Conic from an Equation

Problem: Identify the conic represented by $x^2-4y^2+6x+8y-15=0$.

Solution:

1. Complete the square for $x$ and $y$: $$x^2+6x=(x+3{)}^2-9$$ $$-4y^2+8y=-4(y^2-2y)=-4[(y-1{)}^2-1]=-4(y-1{)}^2+4$$
2. Substitute back: $$(x+3{)}^2-9-4(y-1{)}^2+4-15=0$$ $$(x+3{)}^2-4(y-1{)}^2-20=0$$ $$(x+3{)}^2-4(y-1{)}^2=20$$ $$\frac{(x+3{)}^2}{20}-\frac{(y-1{)}^2}{5}=1$$
3. This is a hyperbola centered at $(-3,1)$ with a horizontal transverse axis.

7. NEB Board-Style Questions

Short Answer Questions

1. Define an ellipse and give its standard equation.
2. What is the eccentricity of a hyperbola? How does it differ from that of an ellipse?
3. Write the equation of the asymptotes for the hyperbola $\frac{(y-2{)}^2}{16}-\frac{(x+1{)}^2}{9}=1$.

Long Answer Questions

1. Derive the standard equation of an ellipse with foci at $(3,0)$ and $(-3,0)$ and a major axis length of 10.
2. Graph the hyperbola $9x^2-16y^2=144$. Identify its center, vertices, foci, and asymptotes.
3. A satellite dish has a cross-section modeled by the hyperbola $y^2-x^2=1$. Find the equations of its asymptotes and sketch the graph.

Problem-Solving Questions

1. Find the distance between the foci of the ellipse $25x^2+16y^2=400$.
2. Determine whether the conic $x^2+4y^2-6x+8y-15=0$ is an ellipse, parabola, or hyperbola. Justify your answer.
3. A planet orbits the Sun in an elliptical path with a semi-major axis of 5 AU and eccentricity of 0.2. Find the distance between the foci of the orbit.

Exam Tip

1. Memorize Standard Forms: Know the standard equations for ellipses and hyperbolas centered at $(h,k)$.
2. Identify Conics Quickly: For any equation, check the signs and coefficients to determine if it’s an ellipse, parabola, or hyperbola.
   • If both $x^2$ and $y^2$ terms are positive and have the same sign: ellipse.
   • If one $x^2$ or $y^2$ term is positive and the other is negative: hyperbola.
   • If one squared term is missing: parabola.
3. Graphing: Always plot the center, vertices, and asymptotes (for hyperbolas) before sketching the curve.
4. Eccentricity: Remember $e=\frac{c}{a}$ and how it differs for ellipses ($0<e<1$) and hyperbolas ($e>1$).
5. Word Problems: Pay attention to real-world applications like orbits, dishes, and towers. Draw diagrams to visualize the scenario.
6. Practice Completing the Square: Many NEB problems require rewriting equations in standard form, so practice completing the square for both $x$ and $y$ terms.

Good luck with your NEB exam preparation! 🚀