Conic Sections & Circles: Definitions, Equations, Properties & Applications

TAKEAWAYS:

• Conic sections are curves formed by slicing a cone at different angles: parabola (opens upward/downward), ellipse (oval shape), and hyperbola (two open curves).
• The circle is a special case of an ellipse where the two axes are equal, with the standard equation $(x-h{)}^2+(y-k{)}^2=r^2$.
• Key properties include focus, directrix, eccentricity, and axes—each defines the shape’s unique characteristics.
• Graphing conic sections requires identifying the center, vertices, and coefficients from their equations.
• Applications range from satellite dishes (parabolas) to planetary orbits (ellipses) and lens designs (hyperbolas).
• The SEE exam tests equation derivation, graph sketching, and real-world problem-solving using these curves.

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### **1. What Are Conic Sections?**
Conic sections are curves formed when a **plane slices a double cone** at different angles. There are **four types**, but we focus on three in this unit:


- **Circle**: Slice is **perpendicular** to the cone’s axis (eccentricity \(e = 0\)).
- **Ellipse**: Slice is **tilted but not too steep** (\(0 < e < 1\)).
- **Parabola**: Slice is **parallel** to the cone’s side (\(e = 1\)).
- **Hyperbola**: Slice is **steeper** than the cone’s side (\(e > 1\)).

**Key Term**:
- **Eccentricity (\(e\))**: Measures how "stretched" the curve is.
  - \(e = 0\) → Circle
  - \(0 < e < 1\) → Ellipse
  - \(e = 1\) → Parabola
  - \(e > 1\) → Hyperbola

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### **2. The Circle: Standard Equation and Properties**
A **circle** is the set of all points in a plane that are **equidistant** from a fixed point (the **center**).

```figure
{"type":"circle","center":{"x":0,"y":0},"radius":5,"points":[{"x":0,"y":5,"label":"r"},{"x":5,"y":0,"label":"r"},{"x":0,"y":-5,"label":"r"},{"x":-5,"y":0,"label":"r"}],"caption":"Circle with center (0,0) and radius 5"}

Standard Equation:

For a circle with:

• Center at ((h, k)),
• Radius (r), the equation is: [ (x - h)^2 + (y - k)^2 = r^2 ]

Example 1: Write the equation of a circle with center ((3, -2)) and radius (5). Solution: Substitute (h = 3), (k = -2), (r = 5): [ (x - 3)^2 + (y - (-2))^2 = 5^2 \ (x - 3)^2 + (y + 2)^2 = 25 ]

Graphing a Circle:

1. Plot the center ((h, k)).
2. From the center, mark points (r) units left, right, up, and down.
3. Draw a smooth curve through these points.

Key Properties:

• Diameter = $2r$
• Circumference = $2\pi r$
• Area = $\pi r^2$

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3. The Parabola: Standard Equations and Graphs

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).

Standard Forms:

1. Vertical Parabola (opens up/down): $$y=a(x-h{)}^2+k$$

   • Vertex: $(h,k)$
   • Focus: $(h,k+\frac{1}{4a})$
   • Directrix: $y=k-\frac{1}{4a}$

2. Horizontal Parabola (opens left/right): $$x=a(y-k{)}^2+h$$

   • Vertex: $(h,k)$
   • Focus: $(h+\frac{1}{4a},k)$
   • Directrix: $x=h-\frac{1}{4a}$

Example 2: Graph $y=2(x-1{)}^2+3$. Identify the vertex, focus, and directrix. Solution:

• Vertex: $(1,3)$
• Focus: $(1,3+\frac{1}{4\times 2})=(1,3.125)$
• Directrix: $y=3-0.25=2.75$

Real-World Use:

• Satellite dishes (focus at the receiver).
• Headlights (reflect light in one direction).

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4. The Ellipse: Standard Equation and Properties

An ellipse is the set of all points where the sum of distances to two fixed points (foci) is constant.

Standard Equation:

$$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$$

• Center: $(h,k)$
• Major axis (longer): $2a$ (horizontal if $a>b$)
• Minor axis (shorter): $2b$ (vertical if $a>b$)
• Foci: $(\pm c,0)$, where $c^2=a^2-b^2$

Example 3: Write the equation of an ellipse with:

• Center $(0,0)$,
• Major axis $10$ (horizontal),
• Minor axis $6$. Solution:
• $a=5$, $b=3$
• $c^2=25-9=16$ → $c=4$
• Equation: $$\frac{x^2}{25}+\frac{y^2}{9}=1$$

Real-World Use:

• Planetary orbits (Earth’s orbit is nearly elliptical).
• Architecture (domes, stadiums).

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5. The Hyperbola: Standard Equation and Properties

A hyperbola is the set of all points where the difference of distances to two fixed points (foci) is constant.

Standard Equations:

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

   • Transverse axis: $2a$ (horizontal)
   • Foci: $(\pm c,0)$, where $c^2=a^2+b^2$

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

   • Transverse axis: $2a$ (vertical)

Example 4: Graph $\frac{x^2}{9}-\frac{y^2}{16}=1$. Find the foci. Solution:

• $a^2=9$ → $a=3$
• $b^2=16$ → $b=4$
• $c^2=9+16=25$ → $c=5$
• Foci: $(\pm 5,0)$

Real-World Use:

• Lens design (telescopes).
• Navigation systems (GPS uses hyperbolic paths).

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6. Comparing Conic Sections

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7. Solved Problems (SEE-Style)

Problem 1:

Find the equation of a circle with diameter endpoints $(-2,3)$ and $(4,-1)$. Solution:

1. Find center (midpoint): $$h=\frac{-2+4}{2}=1,k=\frac{3+(-1)}{2}=1$$
2. Find radius (half the distance between endpoints): $$r=\frac{\sqrt{(4-(-2){)}^2+(-1-3{)}^2}}{2}=\frac{\sqrt{36+16}}{2}=\frac{\sqrt{52}}{2}=\sqrt{13}$$
3. Equation: $$(x-1{)}^2+(y-1{)}^2=13$$

Problem 2:

A parabola has its vertex at $(2,-3)$ and focus at $(2,0)$. Write its equation. Solution:

1. Since the focus is above the vertex, the parabola opens upward.
2. Use the form $y=a(x-h{)}^2+k$.
3. Distance from vertex to focus = $3$ → $\frac{1}{4a}=3$ → $a=\frac{1}{12}$.
4. Equation: $$y=\frac{1}{12}(x-2{)}^2-3$$

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Exam Tip

1. Memorize standard equations for circles, parabolas, ellipses, and hyperbolas.
2. Graphing: Always plot the vertex, center, or foci first.
3. SEE questions often ask:
   • Deriving equations from given properties.
   • Identifying conic sections from graphs.
   • Real-world applications (e.g., "A satellite dish is shaped like a parabola...").
4. Common mistakes:
   • Mixing up $a$ and $b$ in ellipses/hyperbolas.
   • Forgetting to square the radius in circle equations.
   • Misidentifying the transverse axis in hyperbolas.

Practice: Sketch at least 5 conic sections from their equations before the exam!

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