Conic Sections: Circles, Ellipses, Parabolas, Hyperbolas

TAKEAWAYS:

• Conic sections arise from slicing a cone at different angles, yielding circles, ellipses, parabolas, and hyperbolas, each with unique algebraic and geometric properties.
• Standard equations of conic sections (e.g., $\frac{(x-h{)}^2}{a^2}\pm \frac{(y-k{)}^2}{b^2}=1$) define their shape, center, axes, and vertices.
• Graphical transformations (shifts, stretches) modify conic sections while preserving their fundamental properties.
• Applications span satellite orbits (ellipses), parabolic antennas, hyperbolic navigation (LORAN), and computer-generated 3D shapes.
• Polar equations (e.g., $r=\frac{ed}{1+e\cos \theta }$) unify conic sections under a single parameter $e$ (eccentricity).
• Exam questions often test identifying conics from equations, graphing, or solving applied problems (e.g., projectile trajectories).

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1. Introduction to Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone (two cones apex-to-apex). The shape depends on the angle of intersection:

• Circle: Plane perpendicular to the cone’s axis.
• Ellipse: Plane at a shallow angle (cuts one nappe).
• Parabola: Plane parallel to the cone’s side.
• Hyperbola: Plane at a steep angle (cuts both nappes).

Key Idea: All conic sections can be derived from the general second-degree equation: $$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$$ The discriminant $B^2-4AC$ classifies the conic:

• $B^2-4AC<0$: Circle/ellipse
• $B^2-4AC=0$: Parabola
• $B^2-4AC>0$: Hyperbola

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2. Standard Equations and Graphs

Each conic has a standard form centered at $(h,k)$, derived from its geometric properties.

A. Circle

• Equation: $(x-h{)}^2+(y-k{)}^2=r^2$
• Graph: All points equidistant from center $(h,k)$.
• Example: A satellite dish (parabolic) focuses signals to a circular receiver.

Worked Example 1: Find the equation of a circle with diameter endpoints $(1,5)$ and $(5,1)$. Solution:

1. Center $(h,k)$ is midpoint: $h=\frac{1+5}{2}=3$, $k=\frac{5+1}{2}=3$.
2. Radius $r=\frac{\text{distance}}{2}=\frac{\sqrt{(5-1{)}^2+(1-5{)}^2}}{2}=\frac{\sqrt{32}}{2}=2\sqrt{2}$.
3. Equation: $(x-3{)}^2+(y-3{)}^2=(2\sqrt{2}{)}^2$ → $(x-3{)}^2+(y-3{)}^2=8$.

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B. Ellipse

• Equation: $\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$ (major axis horizontal if $a>b$).
• Graph: Sum of distances from any point to two foci is constant.
• Eccentricity: $e=\frac{c}{a}$, where $c^2=a^2-b^2$.

Worked Example 2: Identify the conic and sketch: $4x^2+9y^2-8x+36y+4=0$. Solution:

1. Rewrite: $4(x^2-2x)+9(y^2+4y)=-4$.
2. Complete the square: $$4[(x-1{)}^2-1]+9[(y+2{)}^2-4]=-4⟹4(x-1{)}^2+9(y+2{)}^2=32.$$
3. Divide by 32: $\frac{(x-1{)}^2}{8}+\frac{(y+2{)}^2}{32/9}=1$. Conclusion: Ellipse centered at $(1,-2)$, $a^2=8$, $b^2=32/9$.

Real-World Tie-In:

• Nepal Electricity Authority (NEA) Transmission Lines: Elliptical orbits model power satellite paths (e.g., geostationary satellites) to optimize signal coverage.

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C. Parabola

• Equation: $y=a(x-h{)}^2+k$ (vertical) or $x=a(y-k{)}^2+h$ (horizontal).
• Graph: All points equidistant from focus and directrix.
• Standard Form: $(y-k{)}^2=4p(x-h)$ (opens right/left) or $(x-h{)}^2=4p(y-k)$ (opens up/down).

Worked Example 3: Find the focus and directrix of $x^2-6x+8y+17=0$. Solution:

1. Rewrite: $x^2-6x=-8y-17$.
2. Complete the square: $(x^2-6x+9)=-8y-8$ → $(x-3{)}^2=-8(y+1)$.
3. Compare to $(x-h{)}^2=4p(y-k)$: $4p=-8$ → $p=-2$. Focus: $(h,k+p)=(3,-1-2)=(3,-3)$. Directrix: $y=k-p=-1-(-2)=1$.

Real-World Tie-In:

• Pathao’s Ride Matching Algorithm: Uses parabolic cost functions to minimize driver-passenger pairing time based on location (latitude/longitude treated as parabola parameters).

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D. Hyperbola

• Equation: $\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1$ (horizontal) or $\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1$ (vertical).
• Graph: Difference of distances to two foci is constant.
• Asymptotes: Lines $y=\pm \frac{b}{a}x$ (for centered hyperbolas).

Worked Example 4: Find the equation of a hyperbola with vertices at $(2,5)$ and $(2,1)$, and asymptotes $y=5\pm \frac{3}{4}(x-2)$. Solution:

1. Center $(h,k)=(2,3)$ (midpoint of vertices).
2. Vertical hyperbola: $\frac{(y-3{)}^2}{a^2}-\frac{(x-2{)}^2}{b^2}=1$.
3. $a=2$ (distance from center to vertex).
4. Asymptote slope $\frac{a}{b}=\frac{3}{4}$ → $b=\frac{4}{3}a=\frac{8}{3}$.
5. Equation: $\frac{(y-3{)}^2}{4}-\frac{(x-2{)}^2}{64/9}=1$.

Real-World Tie-In:

• Ncell’s Signal Towers: Hyperbolic navigation (like LORAN) triangulates user location using time delays between towers, modeled by hyperbolas.

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3. Polar Equations of Conics

All conic sections can be expressed in polar coordinates as: $$r=\frac{ed}{1+e\cos \theta }$$ where:

• $e$: eccentricity ($0<e<1$: ellipse; $e=1$: parabola; $e>1$: hyperbola).
• $d$: distance from focus to directrix.

Graphs of $r=\frac{ed}{1+e\cos \theta }$ for $e=0.5$ (ellipse), $e=1$ (parabola), $e=2$ (hyperbola), with focus at origin. (Image: Kent Thele, CC BY-SA 4.0, via Wikimedia Commons)

Worked Example 5: Convert $r=\frac{6}{3+2\cos \theta }$ to Cartesian form. Solution:

1. Rewrite: $r(3+2\cos \theta )=6$ → $3r+2r\cos \theta =6$.
2. Substitute $r=\sqrt{x^2+y^2}$, $r\cos \theta =x$: $$3\sqrt{x^2+y^2}+2x=6.$$
3. Isolate square root and square both sides: $$9(x^2+y^2)=36-24x+4x^2⟹5x^2+9y^2+24x-36=0.$$ Conclusion: Ellipse (since $B^2-4AC=0-4(5)(9)<0$).

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4. Applications in Engineering

Real-World Example:

• Khalti’s Transaction Validation: Uses elliptic curves (a type of conic) for cryptographic security in digital payments.

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5. Comparison Table

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6. Transformations of Conics

Conics can be shifted, stretched, or rotated:

• Shifts: Replace $x$ with $(x-h)$, $y$ with $(y-k)$.
• Stretches: Multiply $x$ or $y$ by a factor (e.g., $a$ or $b$).
• Rotations: Use rotation formulas $x=x^{\prime }\cos \theta -y^{\prime }\sin \theta$, $y=x^{\prime }\sin \theta +y^{\prime }\cos \theta$.

Worked Example 6: Rotate the ellipse $x^2+4xy+y^2=4$ to eliminate the $xy$ term. Solution:

1. Angle $\theta$ satisfies $\cot (2\theta )=\frac{A-C}{B}=\frac{1-1}{4}=0$ → $\theta =45^{\circ }$.
2. Rotation formulas: $$x=\frac{x^{\prime }-y^{\prime }}{\sqrt{2}},y=\frac{x^{\prime }+y^{\prime }}{\sqrt{2}}.$$
3. Substitute into the equation and simplify to get the rotated conic.

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In the Real World

1. eSewa’s Payment Routing: Uses parabolic optimization to route transactions between banks and users, minimizing latency. The cost function $C(d)=a{d}^2+bd$ (where $d$ is distance) ensures faster processing for nearby transactions.

2. Daraz’s Inventory Management: Stores use elliptical distribution models to predict demand spikes (e.g., Diwali season). The equation $\frac{(t-15{)}^2}{25}+\frac{(d-100{)}^2}{100}=1$ models time $t$ vs. demand $d$, where $t=15$ is peak week.

3. NTC’s Power Grid Stability: Hyperbolic functions model voltage fluctuations in long-distance transmission lines. The equation $V=\frac{1}{x}$ (inverse relationship) ensures transformers compensate for drops over distance $x$.

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Exam Tip

1. Identify Conics Quickly:

   • Memorize the discriminant $B^2-4AC$ for the general form.
   • For standard forms, check for $+$ (ellipse/circle), $-$ (hyperbola), or single squared term (parabola).

2. Graphing:

   • Always plot the center $(h,k)$, vertices, and asymptotes (for hyperbolas).
   • Label axes and key points (e.g., foci, directrix).

3. Applications:

   • Link equations to real scenarios (e.g., "This parabola models a satellite dish").
   • For polar forms, recognize $e$ determines the conic type.

4. Common Mistakes:

   • Forgetting to complete the square before identifying conics.
   • Misplacing signs in standard forms (e.g., $-$ for hyperbolas).
   • Confusing $a$ and $b$ in ellipses (always $a$ is the larger denominator).

5. Practice:

   • Convert between Cartesian and polar forms.
   • Solve for missing parameters (e.g., find $p$ in a parabola given focus/directrix).
   • Sketch conics from equations and vice versa.

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Visual Summary: Side-by-side graphs of circle, ellipse, parabola, and hyperbola with axes, foci, and key points labeled. (Image: Original: Magister Mathematicae Derivative work: Phancy Phys, CC BY-SA 3.0, via Wikimedia Commons)