Conic Sections: Ellipses & Hyperbolas – Equations, Properties & Real-World Uses

TAKEAWAYS:

• Ellipses are stretched circles defined by the sum of distances to two foci, with standard form $\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$ and eccentricity $e=\frac{c}{a}$ where $c^2=a^2-b^2$.
• Hyperbolas are "open" curves defined by the difference of distances to two foci, with standard forms $\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1$ (horizontal) or $\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1$ (vertical), and $e=\frac{c}{a}$ where $c^2=a^2+b^2$.
• Key properties of both conics include vertices, foci, directrices, and latus rectum (chord through a focus perpendicular to the major axis), all derivable from their standard equations.
• Real-world applications span satellite orbits (ellipses), antenna design (hyperbolas), and financial modeling (e.g., loan amortization curves).
• Exam focus: Memorize standard forms, derive properties from equations, and recognize transformations (shifts, stretches) in conic equations.
• Common pitfalls: Mixing up $a$ and $b$ for ellipses/hyperbolas, misapplying eccentricity formulas, and forgetting to complete the square for non-standard equations.

1. Introduction to Conic Sections

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

• Ellipse: Plane cuts all generators of one nappe (closed curve).
• Hyperbola: Plane cuts both nappes (open curve).
• Parabola: Plane cuts parallel to one generator (open curve, not covered here).

2. Ellipses: Definition and Standard Form

An ellipse is the set of all points $(x,y)$ such that the sum of distances to two fixed points (foci) is constant. Standard form (center at $(h,k)$): $$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$$ where:

• $a>b$: major axis horizontal.
• $b>a$: major axis vertical.
• Vertices: $(h\pm a,k)$ or $(h,k\pm a)$.
• Foci: $(h\pm c,k)$ or $(h,k\pm c)$, where $c^2=a^2-b^2$.
• Eccentricity: $e=\frac{c}{a}$ ($0<e<1$).
• Directrices: $x=h\pm \frac{a}{e}$ or $y=k\pm \frac{a}{e}$.
• Latus rectum: Length $=\frac{2b^2}{a}$.

Worked Example 1: Rewrite and Identify Properties

Problem: Find the coordinates of vertices, foci, eccentricity, and directrices of the ellipse: $$9x^2+4y^2-18x-16y-11=0$$

Solution:

1. Complete the square: $$\begin{gathered} 9(x^2-2x)+4(y^2-4y)=11 \\ 9[(x-1{)}^2-1]+4[(y-2{)}^2-4]=11 \\ 9(x-1{)}^2+4(y-2{)}^2-9-16=11 \\ 9(x-1{)}^2+4(y-2{)}^2=36 \\ \frac{(x-1{)}^2}{4}+\frac{(y-2{)}^2}{9}=1 \end{gathered}$$ Here, $a^2=9\Rightarrow a=3$, $b^2=4\Rightarrow b=2$, and $b>a$ (vertical major axis).

2. Properties:

   • Vertices: $(1,2\pm 3)=(1,5)$ and $(1,-1)$.
   • Foci: $c^2=a^2-b^2=9-4=5\Rightarrow c=\sqrt{5}$. Foci at $(1,2\pm \sqrt{5})$.
   • Eccentricity: $e=\frac{c}{a}=\frac{\sqrt{5}}{3}$.
   • Directrices: $y=k\pm \frac{a}{e}=2\pm \frac{3}{\sqrt{5}/3}=2\pm \frac{9}{\sqrt{5}}$.

3. Hyperbolas: Definition and Standard Form

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

1. Horizontal transverse axis: $$\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1$$

   • Vertices: $(h\pm a,k)$.
   • Foci: $(h\pm c,k)$, where $c^2=a^2+b^2$.
   • Eccentricity: $e=\frac{c}{a}>1$.
   • Directrices: $x=h\pm \frac{a}{e}$.
   • Asymptotes: $y-k=\pm \frac{b}{a}(x-h)$.

2. Vertical transverse axis: $$\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1$$ (Swap $x$ and $y$ roles in properties above.)

Worked Example 2: Derive Hyperbola Equation from Focus and Directrix

Problem: Find the equation of a hyperbola with focus $(-2,0)$ and directrix $x=-\frac{1}{2}$.

Solution:

1. Definition: For any point $(x,y)$ on the hyperbola, $$\left|\sqrt{(x+2{)}^2+y^2}\right|-\left|x+\frac{1}{2}\right|=2a$$ (Here, $2a$ is the constant difference, but we’ll derive it.)

2. Use eccentricity: For hyperbolas, $e=\frac{c}{a}>1$. The distance from the center $(h,k)$ to the focus is $c$, and to the directrix is $\frac{a}{e}$.

   • Let the center be $(h,0)$ (since focus and directrix are horizontal).
   • Distance from center to focus: $|h-(-2)|=c$.
   • Distance from center to directrix: $|h-(-\frac{1}{2})|=\frac{a}{e}$.
   • For hyperbolas, $e=\frac{c}{a}$, so $\frac{a}{e}=\frac{a^2}{c}$.
   • Thus, $|h+\frac{1}{2}|=\frac{a^2}{c}$.

3. Assume center at origin for simplicity (shift later if needed): Let $h=0$:

   • Focus at $(-2,0)\Rightarrow c=2$.
   • Directrix $x=-\frac{1}{2}\Rightarrow \frac{a}{e}=\frac{1}{2}$.
   • But $e=\frac{c}{a}=\frac{2}{a}$, so $\frac{a}{2/a}=\frac{a^2}{2}=\frac{1}{2}\Rightarrow a^2=1\Rightarrow a=1$.
   • Then $e=2$, $c=2$ (consistent).

4. Standard form: Since the transverse axis is horizontal (focus on x-axis), $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\text{where}{c}^2=a^2+b^2\Rightarrow b^2=c^2-a^2=4-1=3.$$ Thus, the equation is: $$\frac{x^2}{1}-\frac{y^2}{3}=1.$$

4. Comparison Table: Ellipses vs. Hyperbolas

5. Applications in the Real World

Conic sections model phenomena in technology, finance, and nature. Here’s how Nepalese and global companies use them:

Example 1: Satellite Orbits (Ellipses)

• Application: Most satellites (including those for Ncell or NTC) follow elliptical orbits around Earth. The eccentricity determines how "stretched" the orbit is.
• How it works:
  • Earth is at one focus of the ellipse.
  • The apogee (farthest point) and perigee (closest point) correspond to the vertices.
  • Example: A satellite with $a=7000$ km, $b=6900$ km, and Earth’s radius $c=6371$ km (approximate) would have: $$c^2=a^2-b^2\Rightarrow 6371^2=7000^2-6900^2\text{(Check: }40,580,641\approx 49,000,000-47,610,000\text{)}$$ This ensures the satellite’s path is an ellipse with Earth at one focus.

Example 2: GPS and Hyperbolic Navigation

• Application: Systems like Google Maps or Pathao’s delivery routing use hyperbolic principles for time-difference-of-arrival (TDOA) navigation. Two transmitters send signals; the difference in arrival times defines a hyperbola, intersecting with another hyperbola from a third transmitter to pinpoint location.
• How it works:
  • Suppose three cell towers (A, B, C) transmit signals. A user’s phone measures:
    • $|d_A-d_B|=2a$ (hyperbola 1).
    • $|d_B-d_C|=2a^{\prime }$ (hyperbola 2).
  • The intersection of these hyperbolas gives the user’s location.
  • Example: If $a=5$ km and $a^{\prime }=3$ km, the user lies at the intersection of two hyperbolas centered at the towers.

flowchart TD
    A["Tower A"] -->|"Signal"| B["User's Phone"]
    B -->|"Signal"| C["Tower B"]
    B -->|"Signal"| D["Tower C"]
    subgraph Hyperbola1["|d_A - d_B| = 10 km"]
        A -->|"Hyperbola branch 1"| B
        A -->|"Hyperbola branch 2"| B
    end
    subgraph Hyperbola2["|d_B - d_C| = 6 km"]
        B -->|"Hyperbola branch 1"| D
        B -->|"Hyperbola branch 2"| D
    end
    label: "Hyperbolic navigation in GPS"

Example 3: Loan Amortization (Ellipses in Finance)

• Application: Banks (e.g., Nabil Bank, Global IME) use elliptical curves to model loan repayment schedules. The eccentricity of the curve represents the "stretch" between principal and interest payments.
• How it works:
  • A loan’s total repayment can be visualized as an ellipse where:
    • Major axis: Total loan amount + interest.
    • Minor axis: Time (months/years).
  • Example: A ₹1,000,000 loan at 10% annual interest over 5 years (60 months) with equal monthly payments:
    • Total repayment ≈ ₹1,000,000 + ₹50,000 (interest) = ₹1,050,000.
    • The ellipse’s equation (simplified) might be: $$\frac{x^2}{1,050,000^2}+\frac{y^2}{60^2}=1$$ where $x$ = principal + interest at time $y$ (months).

Example 4: Traffic Flow Optimization (Hyperbolas)

• Application: Kathmandu’s traffic management systems (e.g., for buses or Pathao drivers) use hyperbolic paths to optimize routes. Hyperbolas model time-delay contours where the difference in travel time between two paths is constant.
• How it works:
  • Suppose two routes from Thapathali to Lakshmi Narayan have a fixed time difference (e.g., 5 minutes). The set of points where this holds forms a hyperbola.
  • Example: If Route 1 takes $t_1$ minutes and Route 2 takes $t_2$ minutes, the hyperbola is defined by $|t_1-t_2|=5$. Drivers can choose the faster branch.

flowchart TD
    A["Thapathali"] -->|"Route 1 (t1)"| B["Lakshmi Narayan"]
    A -->|"Route 2 (t2)"| B
    subgraph Hyperbola["|t1 - t2| = 5 min"]
        A -->|"Branch 1"| B
        A -->|"Branch 2"| B
    end
    label: "Hyperbolic traffic time contours"

6. Solving Problems from Past Exams

Worked Example 3: Find Eccentricity and Foci of an Ellipse

Problem: Find the eccentricity and foci of the ellipse: $$25x^2+4y^2=100$$

Solution:

1. Rewrite in standard form: $$\frac{x^2}{4}+\frac{y^2}{25}=1$$ Here, $a^2=25\Rightarrow a=5$, $b^2=4\Rightarrow b=2$, and $a>b$ (vertical major axis).

2. Foci: $$c^2=a^2-b^2=25-4=21\Rightarrow c=\sqrt{21}.$$ Foci at $(0,\pm \sqrt{21})$.

3. Eccentricity: $$e=\frac{c}{a}=\frac{\sqrt{21}}{5}.$$

Worked Example 4: Find Equation of Ellipse Given Latus Rectum and Eccentricity

Problem: Find the equation of the ellipse whose latus rectum is 5 and eccentricity is $\frac{1}{\sqrt{2}}$.

Solution:

1. Latus rectum formula: For ellipses, latus rectum length $=\frac{2b^2}{a}=5$.

2. Eccentricity: $e=\frac{c}{a}=\frac{1}{\sqrt{2}}\Rightarrow c=\frac{a}{\sqrt{2}}$.

3. Relation between $a,b,c$: $c^2=a^2-b^2\Rightarrow {\left(\frac{a}{\sqrt{2}}\right)}^2=a^2-b^2\Rightarrow \frac{a^2}{2}=a^2-b^2\Rightarrow b^2=\frac{a^2}{2}$.

4. Substitute into latus rectum: $\frac{2\cdot \frac{a^2}{2}}{a}=5\Rightarrow a=5$.

5. Find $b$: $b^2=\frac{5^2}{2}=\frac{25}{2}\Rightarrow b=\frac{5\sqrt{2}}{2}$.

6. Equation: Since $a>b$, the major axis is horizontal: $$\frac{x^2}{25}+\frac{y^2}{\frac{25}{2}}=1\text{or}\frac{x^2}{25}+\frac{2y^2}{25}=1.$$

7. Key Formulas Summary

8. Exam Tip

1. Always complete the square for non-standard equations to identify $h,k,a,b$.
2. Memorize standard forms and when to use horizontal vs. vertical axes.
3. For ellipses:
   • If $a>b$, major axis is horizontal; else vertical.
   • $c^2=a^2-b^2$ and $e=\frac{c}{a}$.
4. For hyperbolas:
   • The transverse axis determines the standard form.
   • Asymptotes are critical for sketching; they pass through the center with slopes $\pm \frac{b}{a}$.
5. Common mistakes:
   • Forgetting to take square roots when solving for $a,b,c$.
   • Mixing up $a$ and $b$ for ellipses/hyperbolas.
   • Incorrectly applying eccentricity formulas (e.g., using $e=\frac{a}{c}$ for hyperbolas).
6. Graphical questions:
   • Always sketch the conic, label vertices/foci, and draw asymptotes (for hyperbolas) or directrices.
   • Use symmetry to plot points quickly.

9. Practice Problems

1. Rewrite $16x^2+9y^2-32x+36y-47=0$ in standard form and find its foci.
2. Find the equation of the hyperbola with vertices at $(\pm 3,0)$ and foci at $(\pm 5,0)$.
3. A satellite’s orbit has semi-major axis $a=7000$ km and eccentricity $e=0.1$. Find its semi-minor axis $b$ and the distance between foci.
4. For the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, find the length of the latus rectum and the equations of the directrices.
5. Derive the equation of the hyperbola with focus $(4,0)$ and directrix $x=1$.

10. Final Notes

• Ellipses are used in astronomy (planetary orbits), architecture (whispering galleries), and medical imaging (PET scans).
• Hyperbolas appear in navigation (LORAN), antenna design (radio telescopes), and economics (indifference curves).
• Always verify your answers by checking:
  • For ellipses: $c^2=a^2-b^2$ and $e<1$.
  • For hyperbolas: $c^2=a^2+b^2$ and $e>1$.