Comprehensive Pre-Calculus Conic Sections Study Guide

Definitions and Geometric Foundations of Conic Sections

• Circle: A conic section defined as the collection of all points in a plane that are located at a fixed distance, denoted as $r$, from a fixed central point designated as $(h, k)$.

• Ellipse: A conic section defined as the collection of points in a plane such that the sum of the distances from any point on the curve to two fixed points, known as the foci, is a constant value.

• Hyperbola: A conic section defined as the collection of points in a plane such that the absolute difference of the distances from any point on the curve to two fixed points (foci) is a constant value.

• Parabola: A conic section defined as the collection of points in a plane that are equidistant from a fixed point, known as the focus, and a fixed line, known as the directrix.

• Cross Sections of a Cone: Conic sections are formed by the intersection of a plane and a right circular cone. Depending on the angle of the plane, the resulting shapes include the circle, ellipse, parabola, and hyperbola.

Parabolas: Properties and Equations

• Standard Form of Parabolic Equations:     * Vertical Axis of Symmetry: $(x - h)^2 = 4p(y - k)$         * Vertex: $(h, k)$         * Focus: $(h, k + p)$         * Directrix: $y = k - p$         * Axis of Symmetry: $x = h$     * Horizontal Axis of Symmetry: $(y - k)^2 = 4p(x - h)$         * Vertex: $(h, k)$         * Focus: $(h + p, k)$         * Directrix: $x = h - p$         * Axis of Symmetry: $y = k$

• Latus Rectum (LR): The line segment passing through the focus, perpendicular to the axis of symmetry, with endpoints on the parabola. Its total length is given by the absolute value of the coefficient of the linear term, $|4p|$.

• Specific Examples and Solutions:     * Case 1: $x^2 = 20y$         * Here, $4p = 20$, so $p = 5$.         * Vertex: $(0, 0)$         * Focus: $(0, 5)$         * Directrix: $y = -5$         * Axis of Symmetry: $x = 0$         * LR Length: $20$     * Case 2: $(y - 3)^2 = -12(x + 4)$         * Here, $4p = -12$, so $p = -3$.         * Vertex: $(-4, 3)$         * Focus: $(-7, 3)$         * Directrix: $x = -1$         * Axis of Symmetry: $y = 3$         * LR Length: $12$     * Case 3: $(x + 2)^2 = -8(y - 1)$         * Vertex: $(-2, 1)$; Focus: $(-2, -1)$; Directrix: $y = 3$.     * Case 4: $(y - 8)^2 = 48(x - 9)$         * Vertex: $(9, 8)$; Focus: $(21, 8)$; Directrix: $x = -3$.

• Converting General Form to Standard Form:     * Example: $y^2 - 6y - 4x + 29 = 0$         * Rearrange: $y^2 - 6y = 4x - 29$         * Complete the square: $y^2 - 6y + 9 = 4x - 29 + 9$         * Result: $(y - 3)^2 = 4(x - 5)$     * Example: $x^2 + 14x - 8y + 9 = 0$         * Rearrange: $x^2 + 14x = 8y - 9$         * Complete the square: $x^2 + 14x + 49 = 8y - 9 + 49$         * Result: $(x + 7)^2 = 8(y + 5)$

Ellipses: Properties and Calculations

• Standard Form of Ellipse Equations:     * Horizontal Major Axis: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, where a > b.     * Vertical Major Axis: $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$, where a > b.

• Key Relationships:     * Length of major axis: $2a$.     * Length of minor axis: $2b$.     * Distance from center to foci ($c$): Satisfies the equation $c^2 = a^2 - b^2$.

• Specific Examples and Solutions:     * Foci $(-3, 0), (3, 0)$ and Vertices $(-5, 0), (5, 0)$:         * Center is $(0, 0)$. Since foci and vertices are on the x-axis, it is horizontal.         * $a = 5$, $c = 3$.         * $b^2 = a^2 - c^2 = 5^2 - 3^2 = 16$.         * Equation: $\frac{x^2}{25} + \frac{y^2}{16} = 1$     * Foci $(0, -3), (0, 3)$ and Vertices $(0, -7), (0, 7)$:         * Center $(0, 0)$, vertical orientation.         * $a = 7$, $c = 3$.         * $b^2 = 7^2 - 3^2 = 49 - 9 = 40$.         * Equation: $\frac{x^2}{40} + \frac{y^2}{49} = 1$     * Major axis horizontal length 18; Minor axis length 12; Center $(0, 0)$:         * $2a = 18 \rightarrow a = 9$         * $2b = 12 \rightarrow b = 6$         * Equation: $\frac{x^2}{81} + \frac{y^2}{36} = 1$     * Major axis vertical length 16; Minor axis length 8; Center $(-3, 5)$:         * $2a = 16 \rightarrow a = 8$         * $2b = 8 \rightarrow b = 4$         * Equation: $\frac{(x + 3)^2}{16} + \frac{(y - 5)^2}{64} = 1$

Hyperbolas: Properties and Asymptotes

• Standard Form of Hyperbola Equations:     * Horizontal Transverse Axis: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$     * Vertical Transverse Axis: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$

• Key Relationships:     * Distance from center to foci ($c$): Satisfies the equation $c^2 = a^2 + b^2$.     * Asymptotes (Horizontal): $y - k = \pm \frac{b}{a}(x - h)$     * Asymptotes (Vertical): $y - k = \pm \frac{a}{b}(x - h)$

• Specific Examples and Solutions:     * Foci $(0, -10), (0, 10)$ and Vertices $(0, -5), (0, 5)$:         * Vertical orientation; Center $(0, 0)$.         * $a = 5$, $c = 10$.         * $b^2 = c^2 - a^2 = 10^2 - 5^2 = 75$.         * Equation: $\frac{y^2}{25} - \frac{x^2}{75} = 1$     * Finding Asymptotes for $\frac{(x - 4)^2}{16} - \frac{(y + 1)^2}{36} = 1$:         * Horizontal hyperbola; $a = 4$, $b = 6$.         * Slopes: $\pm \frac{b}{a} = \pm \frac{6}{4} = \pm \frac{3}{2}$.         * Equations: $y + 1 = \pm \frac{3}{2}(x - 4)$.     * Equation given Asymptotes $(y - 9) = \pm \frac{9}{11}(x + 5)$ and vertical transverse axis:         * Here $a = 9$ and $b = 11$.         * Equation: $\frac{(y - 9)^2}{81} - \frac{(x + 5)^2}{121} = 1$

General Conic Transformations and Applications

• Completing the Square for Mixed Conics:     * Circle Example: $x^2 + y^2 - 10x + 6y + 18 = 0$         * $(x^2 - 10x + 25) + (y^2 + 6y + 9) = -18 + 25 + 9$         * $(x - 5)^2 + (y + 3)^2 = 16$         * Conic: Circle; Center: $(5, -3)$; Radius: $4$.     * Ellipse Example: $2x^2 + 8y^2 - 8x + 80y + 136 = 0$         * $2(x^2 - 4x) + 8(y^2 + 10y) = -136$         * $2(x^2 - 4x + 4) + 8(y^2 + 10y + 25) = -136 + 8 + 200$         * $2(x - 2)^2 + 8(y + 5)^2 = 72$         * Divide by 72: $\frac{(x - 2)^2}{36} + \frac{(y + 5)^2}{9} = 1$         * Conic: Ellipse; Center: $(2, -5)$; Vertices: $(8, -5), (-4, -5)$; Foci: $(2 \pm 3\sqrt{3}, -5)$.

• Real-World Application: The Suspension Bridge:     * Towers are $800\,ft$ apart and rise $160\,ft$ above the road.     * Cable touches road midway (Vertex at $(0, 0)$).     * Points on parabola: $(\pm 400, 160)$.     * $x^2 = 4py \rightarrow 400^2 = 4p(160) \rightarrow 160000 = 640p \rightarrow p = 250$.     * Equation: $x^2 = 1000y$.     * Height $100\,ft$ from a tower means $x = 400 - 100 = 300$.     * $300^2 = 1000y \rightarrow 90000 = 1000y \rightarrow y = 90\,ft$.

• Real-World Application: Rockville Viaduct:     * Stone-arch railroad viaduct over Susquehanna River with $48$ semicircular arches.     * Span of arch: $70\,ft$ (Radius $= 35\,ft$).     * Arch 1 Equation: $x^2 + y^2 = 35^2$ (for $y \geq 0$).     * Arch 2 Equation (shifted): $(x - 70)^2 + y^2 = 35^2$ (for $y \geq 0$).

• Planetary Eccentricities and Ellipses:     * Eccentricity ($e$): For an ellipse, $e = \frac{c}{a}$. As $e$ approaches $0$, the ellipse becomes more circular.     * Data Table:         * Moon: $e \approx 0.0549$         * Mercury: $e \approx 0.2056$         * Venus: $e \approx 0.0068$         * Earth: $e \approx 0.0167$         * Mars: $e \approx 0.0934$         * Jupiter: $e \approx 0.0484$         * Saturn: $e \approx 0.0542$         * Uranus: $e \approx 0.0472$         * Neptune: $e \approx 0.0086$     * Historical Context: Early astronomers struggled to distinguish planetary orbits from circles because most eccentricities are extremely low (e.g., Venus at $0.0068$ and Neptune at $0.0086$), making the orbits appear nearly perfectly circular to the naked eye or limited instrumentation.