Honors Precalculus Chapter 8 Summary Conic Sections- Parabola

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1 Honors Precalculus Chapter 8 Summary Conic Sections- Parabola Definition: Focal length: y- axis P(x, y) Focal chord: focus Vertex x-axis directrix Focal width/ Latus Rectum: Derivation of equation of parabola: Assume that the vertex V = (0, 0), focus F on the positive y-axis, distance from V to F is a, directrix is y = -a. Let P (x, y) be any point on the parabola. Distance from P to F = Distance from to. Using the distance formula: Square both sides: Expand and simplify: Honors_Precalculus_Ch. 8_Summary pg 1 of 14

2 Parabola with vertex at (h, k) Description Equation Graph Vertex: Focus: Directrix: Axis of symmetry: Opens: Vertex: Focus: Directrix: Axis of symmetry: Opens: Given the general equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, how to identify that it is a parabola: a) If B = 0: b) If B 0 : PRACTICE: 1. Graph and find the equation in standard form for the parabola with focus (2, -1) and directrix x = Stein Glass, Inc., makes parabolic headlights for a variety of automobiles. If one of its headlights has a parabolic surface generated by a parabola x 2 = 12y, where should its light bulb be placed? Honors_Precalculus_Ch. 8_Summary pg 2 of 14

3 Conic Sections- Ellipse Definition: P(x, y) F1 F 2 Derivation of equation of an ellipse: Assume that the center = (0, 0), foci at F 1 = (- c, 0) and F 2 = (c, 0). Let P (x, y) = any point on the ellipse. Distance from P to F 1 + Distance from to = 2a Using the distance formula: Isolate one radical: Square both sides: Expand and simplify: Isolate the radical: Divide both sides by 4: Square both sides again: Rearrange terms & factor: Let b 2 = a 2 c 2 Honors_Precalculus_Ch. 8_Summary pg 3 of 14

4 Ellipse with center at (h, k) Description Equation Graph Center: Vertices: Foci: Major axis: Minor axis: Pythagorean relation: Given the general equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, how to identify that it is an ellipse? circle? a) If B = 0: b) If B 0: PRACTICE: 1. Find the equation in standard form for the ellipse with foci (1, -1) and (5, -1) and the major axis with endpoints (0, -1) and (6, -1) A semielliptical archway over a one-way road has a height of 110 feet and a width of 40 feet. Your truck has a width of 10 feet and a height of 9 feet. Will your truck clear the opening of the archway? Honors_Precalculus_Ch. 8_Summary pg 4 of 14

5 Conic Sections- Hyperbola Definition: 10 5 P(x, y) F 1 V 1 V 2 F Derivation of equation of a hyperbola: Assume that the center = (0, 0), foci at F 1 = (- c, 0) and F 2 = (c, 0). Let P (x, y) = any point on the ellipse. Distance from P to F 1 Distance from to = ± 2a -10 Honors_Precalculus_Ch. 8_Summary pg 5 of 14

6 Hyperbola with center at (h, k) Description Equation Graph Center: Vertices: Foci: Asymptotes: Transverse Axis: Pythagorean relation: Center: Vertices: Foci: Asymptotes: Transverse Axis: Pythagorean relation: Given the general equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 and B = 0, how to identify that it is a hyperbola: a) if B = 0 b) if B 0 Honors_Precalculus_Ch. 8_Summary pg 6 of 14

7 PRACTICE: 1. Find the equation in standard form of the hyperbola with center (4, 2), focus (7, 2), vertex (6, 2). What are the equations of the asymptotes? An explosion is recorded by three microphones as illustrated in the figure. Microphone 1 received the sound 4 seconds before Microphone 2 and Microphone 3 received the sound 3 seconds before Microphone 2. Assuming sound travels at 1100 feet per second, determine the possible location of the explosion relative to the location of the microphones. M 1 2 miles M 3 M miles Honors_Precalculus_Ch. 8_Summary pg 7 of 14

8 ROTATED CONICS 1. Solve for y and use a grapher to graph the conic: xy y 8 = 0 2. Solve for y and use a grapher to graph the conic: 2x 2 xy + 3y 2 3x + 4y 6 = 0 3. Derivation of rotation formulas: Let P(r, ") = polar coordinates of point P with respect to x & y axes. Express x & y in terms of r and". y X = (1) Y = (2) y' r P(x, y) x' Let "= angle between x and x axes, 0 < " < π/2. "! The polar coordinates of P with respect to the new x and y axes are: (, ) x X = r cos( ) = (3) Y = = (4) Honors_Precalculus_Ch. 8_Summary pg 8 of 14

9 Substituting (1) and (2) into (3) and (4) yields X = Y = Solve for x and y in terms of x and y : (5) Substitute the results of (5) into the general equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 To align the coordinate axes with the focal axis of the conic, we eliminate the x y term by setting Honors_Precalculus_Ch. 8_Summary pg 9 of 14

10 Discriminant Test: The second-degree equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 graphs as a hyperbola a parabola an ellipse Example: Identify the type of conic, and rotate the coordinate axes to eliminate the xy term. Write and graph the transformed equation. 3x xy + y 2-14 = 0 Honors_Precalculus_Ch. 8_Summary pg 10 of 14

11 POLAR EQUATIONS OF CONICS Focus Directrix Definition of Conic Section (pg. 676): Directrix D P V F If P is a point of a conic section, F is the conic s focus, and D is the point of the directrix closest to P, then Eccentricity = e = The conic is a hyperbola if, parabola if, ellipse if. To obtain a polar equation for a conic section, position the pole at the conic s focus and the polar axis along the focal axis with the directrix to the right of the pole. In terms of r and ", P (r,!) Directrix D PF = PD =! F PF = e PD becomes x = k Solve for r: Honors_Precalculus_Ch. 8_Summary pg 11 of 14

12 Four standard orientations of conic in polar plane ke ke r = r = 1+ ecos" 1" ecos# ke r = 1+ esin" ke r = 1" esin# PRACTICE: 1. Determine the eccentricity, type of conic, and directrix for the given conic. Graph the conic. 2 r = 1+ cos" 2. Read pg to learn about application of polar equations of conics in orbital motion. Honors_Precalculus_Ch. 8_Summary pg 12 of 14

13 THREE DIMENSIONAL CARTESIAN COORDINATE SYSTEM 2D Plot point (2, 5)) Plot point (1, 2, 5) 3D Distance formula: Midpoint formula: Equation of circle: Equation of sphere: Vectors in 2D: Unit vectors: Dot product of 2 vectors: Vectors in 3D: Unit vectors: Dot product of 2 vectors: Equation of line in vector form: A line through the point P(x 0, y 0 ) in the direction of a nonzero vector v = <a, b> has equation: <x, y> = <x 0, y 0 > + t <a, b>, t = real number Equation of line in vector form: Equation of line in parametric form: Honors_Precalculus_Ch. 8_Summary pg 13 of 14

14 Graph ax + by = c in 2D is a. Ex: a) Graph y = 3. Graph of ax + by + cz = d in 3 D is a. Ex: a) Graph y = 3. b) Graph 2x + y = 5 b) Graph z = -2 c) Graph 2x + y = 5 c) Graph 2x + y 3z = 6 Honors_Precalculus_Ch. 8_Summary pg 14 of 14