Unit 11: Conic Sections and Polar Coordinates

Precalculus Honors | Standard PC.MGSR.1

Brian Risden

Unit Overview

We will analyze the behaviors of conic sections and polar coordinates to model mathematical and real-world situations.

Standard Form (1.1) — Identify and graph circles, parabolas, ellipses, and hyperbolas.
General Form & Applications (1.2) — Complete the square to convert equations and investigate real-world models like orbital paths and satellite dishes.
Polar Coordinates (1.3) — Define polar coordinates and relate them to the Cartesian plane.

Conic Sections in Standard Form

The Four Conics

Each conic section is defined by its standard form equation centered at \((h, k)\):

Conic	Standard Form Equation
Circle	\((x-h)^2 + (y-k)^2 = r^2\)
Parabola	\((y-k)^2 = 4p(x-h)\) or \((x-h)^2 = 4p(y-k)\)
Ellipse	\(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)
Hyperbola	\(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) (Horizontal)

Interactive: Ellipse Explorer

t = Array.from({length: 101}, (_, i) => (i / 100) * 2 * Math.PI)
ellipsePoints = t.map(phi => [h + a * Math.cos(phi), k + b * Math.sin(phi)])

Plot.plot({
  aspectRatio: 1,
  x: {domain: [-12, 12], grid: true},
  y: {domain: [-12, 12], grid: true},
  marks: [
    Plot.line(ellipsePoints, {stroke: "#AA4499", strokeWidth: 3}),
    Plot.dot([[h, k]], {fill: "#332288", r: 4}),
    Plot.text([[h, k - 1]], {text: [`Center: (${h}, ${k})`], fill: "#332288"})
  ]
})

Practice: PC.MGSR.1.1

1. Identify the center and radius of \((x+3)^2 + (y-5)^2 = 49\).

Center: \((-3, 5)\) \(\quad\) Radius: \(7\)

2. Is \(\frac{x^2}{25} - \frac{y^2}{9} = 1\) an ellipse or a hyperbola?

Hyperbola (due to the subtraction).

3. Find the vertex and direction of opening for \((x-2)^2 = -8(y+1)\).

Vertex: \((2, -1)\) \(\quad\) Opens: Down (since \(4p\) is negative).

4. Find the foci of the ellipse \(\frac{x^2}{25} + \frac{y^2}{16} = 1\).

\(c^2 = a^2 - b^2 = 25 - 16 = 9 \rightarrow c=3\) \(\quad\) Foci: \((\pm 3, 0)\)

5. Graph the circle centered at the origin with a diameter of 10.

\(r = 5 \rightarrow x^2 + y^2 = 25\)

General Form & Real-World Modeling

Converting to Standard Form

General Form: \(Ax^2 + Cy^2 + Dx + Ey + F = 0\). To graph, we complete the square:

Group \(x\) and \(y\) terms; move constants to the right.
Factor out leading coefficients from \(x^2\) and \(y^2\).
Add \((\frac{b}{2})^2\) to both sides.
Factor into \((x-h)^2\) and \((y-k)^2\).

Real-World Conics

Conic sections model critical physical behaviors:

Parabolas: Satellite dishes and flashlights (reflective properties).
Ellipses: Planetary orbital paths and “whispering galleries.”
Hyperbolas: Sonic booms and certain long-range navigation systems (LORAN).

Practice: PC.MGSR.1.2

1. Convert \(x^2 + y^2 - 4x + 6y - 3 = 0\) to standard form.

\((x^2 - 4x + 4) + (y^2 + 6y + 9) = 3 + 4 + 9\) \(\rightarrow\) \((x-2)^2 + (y+3)^2 = 16\)

2. A satellite dish is 10ft wide and 2ft deep. Where is the focus?

Model as \(x^2 = 4py\). Point \((5, 2)\) is on the curve. \(25 = 4p(2) \rightarrow 8p = 25 \rightarrow p = 3.125\) \(\quad\) Focus: 3.125 ft above the vertex.

3. Identify the conic: \(4x^2 - 9y^2 + 32x + 36y - 8 = 0\).

Hyperbola (coefficients of \(x^2\) and \(y^2\) have opposite signs).

4. A whispering gallery is an elliptical room. If the room is 100ft long and 60ft wide, where should two people stand to hear each other?

Stand at the foci. \(a=50, b=30 \rightarrow c^2 = 2500 - 900 = 1600\). \(c=40\). They should stand 40ft from the center on either side.

5. Convert \(y^2 - 8x - 4y + 12 = 0\) to standard form.

\((y^2 - 4y + 4) = 8x - 12 + 4\) \(\rightarrow\) \((y-2)^2 = 8(x-1)\)

Polar Coordinates

Polar vs. Cartesian

Polar coordinates represent a point as \((r, \theta)\), where \(r\) is the distance from the pole and \(\theta\) is the angle.

Conversion Formulas:

To Cartesian (\(x, y\))	To Polar (\(r, \theta\))
\(x = r \cos \theta\)	\(r^2 = x^2 + y^2\)
\(y = r \sin \theta\)	\(\tan \theta = \frac{y}{x}\)

Interactive: Polar Plotter

pol_rad = pol_deg * Math.PI / 180
pol_x = pol_r * Math.cos(pol_rad)
pol_y = pol_r * Math.sin(pol_rad)

Plot.plot({
  aspectRatio: 1,
  x: {domain: [-11, 11], grid: true},
  y: {domain: [-11, 11], grid: true},
  marks: [
    // Circles for polar grid
    [2, 4, 6, 8, 10].map(d => Plot.dot([[0,0]], {r: d * 20, stroke: "#eee", fill: "none"})),
    Plot.dot([[pol_x, pol_y]], {fill: "#44AA99", r: 6}),
    Plot.ruleX([0], {stroke: "#ccc"}),
    Plot.ruleY([0], {stroke: "#ccc"}),
    Plot.text([[pol_x, pol_y + 1]], {text: [`(${pol_r}, ${pol_deg}°) → (${pol_x.toFixed(2)}, ${pol_y.toFixed(2)})`], fill: "#44AA99", fontSize: 20})
  ]
})

Practice: PC.MGSR.1.3

1. Convert \((4, \frac{\pi}{6})\) to Cartesian coordinates.

\(x = 4 \cos(\pi/6) = 2\sqrt{3}, \quad y = 4 \sin(\pi/6) = 2\) \(\rightarrow\) \((2\sqrt{3}, 2)\)

2. Convert \((-3, -3)\) to polar coordinates (using radians).

\(r = \sqrt{9+9} = 3\sqrt{2}\) \(\quad\) \(\theta = \tan^{-1}(1)\) in Q3 \(\rightarrow 5\pi/4\) \((3\sqrt{2}, 5\pi/4)\)

3. Convert the rectangular equation \(x^2 + y^2 = 25\) to polar form.

\(r^2 = 25 \rightarrow r = 5\)

4. Convert \(r = 3 \csc \theta\) to rectangular form.

\(r = \frac{3}{\sin \theta} \rightarrow r \sin \theta = 3\) \(\rightarrow\) \(y = 3\)

5. Identify the shape of the polar equation \(r = 4 \cos \theta\).

\(r^2 = 4r \cos \theta \rightarrow x^2 + y^2 = 4x\) \((x-2)^2 + y^2 = 4\) \(\rightarrow\) Circle centered at \((2, 0)\) with radius 2.

Summary of Models

Conic/System	Real-World Context	Key Logic
Parabola	Satellite Dishes	Reflection to a focus
Ellipse	Orbits	Two-point constant distance
Hyperbola	Navigation	Constant difference distance
Polar	Radar/Navigation	Distance and angle from center

Conic	Standard Form Equation
Circle	\((x-h)^2 + (y-k)^2 = r^2\)
Parabola	\((y-k)^2 = 4p(x-h)\) or \((x-h)^2 = 4p(y-k)\)
Ellipse	\(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)
Hyperbola	\(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) (Horizontal)