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
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 = 5. 4 2-5 5-2 -4 2. 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
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
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). 4 2-5 5-2 -4 2. 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
Conic Sections- Hyperbola Definition: 10 5 P(x, y) F 1 V 1 V 2 F 2-10 10-5 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
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
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? 4 2-5 5-2 -4 2. 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 2 1.5 miles Honors_Precalculus_Ch. 8_Summary pg 7 of 14
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
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
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 2 + 2 3 xy + y 2-14 = 0 Honors_Precalculus_Ch. 8_Summary pg 10 of 14
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
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 681-682 to learn about application of polar equations of conics in orbital motion. Honors_Precalculus_Ch. 8_Summary pg 12 of 14
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
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