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mathematics
precalculus 1st
Precalculus 1st Edition Jay Abramson - Solutions
For the following exercises, use the given information about the graph of each ellipse to determine its equation.Center (3,5); vertex (3,11); one focus: (3,5 + 4√2)
For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.Directrix is y = −2 and eccentricity e = 4
For the following exercises, given the graph of the hyperbola, find its equation. -10-8-6-4 گیا y 10+ 8+ 6+ -6- 8+ -10- 4 6 8 10 X
For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes.6x2 − 5xy + 6y2 + 20x − y = 0
For the following exercises, given the graph of the hyperbola, find its equation. y (1-√2, 1) Vertices - Foci (1-V5, 1) Center (1, 1) → (1 + √2, 1) Foci (1 + V5, 1) X
For the following exercises, use the given information about the graph of each ellipse to determine its equation.Center (−3,4); vertex (1,4); one focus: (−3 + 2√3,4)
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.Directrix: x = −2; e = 8/3
For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes.6x2 − 8√3 xy + 14y2 + 10x − 3y = 0
For the following exercises, determine the equation for the parabola from its graph. Axis of symmetry Focus (-1,2) Vertex (3,2) X
For the following exercises, given the graph of the ellipse, determine its equation. -6 -4-2 -8 -4 -2 0 -2 -4 -5- 2 +X 46
For the following exercises, given the graph of the hyperbola, find its equation. Vertices (-1,3√2) Foci (-1,3) Center (-1,0) (-1,-3) Foci (-1,-3√2) -x
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.Directrix: x = −5; e = 3/4
For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes.4x2 + 6√3 xy + 10y2 + 20x − 40y = 0
For the following exercises, determine the equation for the parabola from its graph. Vertex (-2,2) y Focus 31 (-1/2) 16 Axis of symmetry
For the following exercises, given the graph of the ellipse, determine its equation. -108 -6 -4 Y -2 4+ 2+ 0 -2- -4- -6 2 46 8 10 X
For the following exercises, determine the equation for the parabola from its graph. Focus 64 Vertex I (-3,5)! Axis of | symmetry y X
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.Directrix: y = 2; e = 2.5
For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes.8x2 + 3xy + 4y2 + 2x − 4 = 0
For the following exercises, given the graph of the hyperbola, find its equation. y Foci (3,1+√7) Center (3, 1). (3,1 + √2) (3,1-√2) (3,1-√7) Foci Vertices
For the following exercises, given the graph of the ellipse, determine its equation. -6-4 -2 -8. -6+ -2 0 -2- -4 -6+ -5 2 +x 4 6
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.Directrix: x = −3; e = 1/3
For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes.16x2 + 24xy + 9y2 + 20x − 44y = 0
For the following exercises, determine the equation for the parabola from its graph. Vertex (-V2, V3) y Focus (-V2+V5, V3) Axis of symmetry -X
For the following exercises, given the graph of the hyperbola, find its equation. Foci (-3-5√2,-3). (-8,-3) (-3,-3) Center y Vertices X Foci • (-3+5√2,-3) (2,-3)
For the following exercises, given the graph of the ellipse, determine its equation. -5 -4 3 -2 5+ 4+ 3+ 2+ 1 0 1
For the following exercises, determine the value of k based on the given equation.Given 4x2 + kxy + 16y2 + 8x + 24y − 48 = 0,find k for the graph to be a parabola.
Recall from Rotation of Axes that equations of conics with an xy term have rotated graphs. For the following exercises, express each equation in polar form with r as a function of θ. xy = 2
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. V(0,0), Endpoints (2,1), (−2,1)
For the following exercises, given the graph of the ellipse, determine its equation. H -3-2 -1 y 5 4 3 لیا 1 12 34 +X 51
For the following exercises, express the equation for the hyperbola as two functions, with y as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.x2/4 − y2/9 = 1
Recall from Rotation of Axes that equations of conics with an xy term have rotated graphs. For the following exercises, express each equation in polar form with r as a function of θ.x2 + xy + y2 = 4
For the following exercises, determine the value of k based on the given equation.Given 2x2 + kxy + 12y2 + 10x − 16y + 28 = 0,find k for the graph to be an ellipse.
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a ⋅ b ⋅ π. (x - 3)² 9 + (y - 3)² 16 = 1
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.V(0,0), Endpoints (−2,4), (−2,−4)
For the following exercises, express the equation for the hyperbola as two functions, with y as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.y2/9 − x2/1 = 1
Recall from Rotation of Axes that equations of conics with an xy term have rotated graphs. For the following exercises, express each equation in polar form with r as a function of θ.2x2 + 4xy + 2y2 = 9
For the following exercises, express the equation for the hyperbola as two functions, with y as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.(x − 2)2/16 − (y + 3)2/25 = 1
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a ⋅ b ⋅ π. (x + 6)² 16 + (y - 6)² 36 = 1
For the following exercises, determine the value of k based on the given equation.Given 3x2 + kxy + 4y2 − 6x + 20y + 128 = 0,find k for the graph to be a hyperbola.
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.V(4,−3), Endpoints (5,−7/2),(3,−7/2)
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a ⋅ b ⋅ π. (x + 1)² 4 + (y-2)² 5 = 1
Recall from Rotation of Axes that equations of conics with an xy term have rotated graphs. For the following exercises, express each equation in polar form with r as a function of θ.16x2 + 24xy + 9y2 = 4
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. V(−3,−1), Endpoints (0,5), (0,−7)
For the following exercises, express the equation for the hyperbola as two functions, with y as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.−4x2 − 16x + y2 − 2y − 19 = 0
For the following exercises, determine the value of k based on the given equation.Given kx2 + 8xy + 8y2 − 12x + 16y + 18 = 0,find k for the graph to be a parabola.
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a ⋅ b ⋅ π.4x2 − 8x + 9y2 − 72y + 112 = 0
Recall from Rotation of Axes that equations of conics with an xy term have rotated graphs. For the following exercises, express each equation in polar form with r as a function of θ.2xy + y = 1
For the following exercises, express the equation for the hyperbola as two functions, with y as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.4x2 − 24x − y2 − 4y + 16 = 0
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.V(4,−3), Endpoints (5,−7/2),(3,−7/2)
For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. The hedge will follow the asymptotes y = x and y = −x, and its closest distance to the center fountain is
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.The mirror in an automobile headlight has a parabolic cross-section with the light bulb at the focus. On a schematic, the equation of the parabola is given as x2=4y. At what
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a ⋅ b ⋅ π.9x2 − 54x + 9y2 − 54y + 81 = 0
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.If we want to construct the mirror from the previous exercise such that the focus is located at (0,0.25), what should the equation of the parabola be?
For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. The hedge will follow the asymptotes y = 2x and y = −2x, and its closest distance to the center fountain
For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.The hedge will follow the asymptotes y = 1/2 x and y = −1/2 x, and its closest distance to the center fountain
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a ⋅ b ⋅ π.Find the equation of the ellipse that will just fit inside a box that is 8 units wide and 4 units high.
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a ⋅ b ⋅ π.Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of h, the height.
For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.The hedge will follow the asymptotes y = 2/3 x and y = − 2/3 x, and its closest distance to the center fountain
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.Consider the satellite dish from the previous exercise. If the dish is 8 feet across at the opening and 2 feet deep, where should we place the receiver?
For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.The hedge will follow the asymptotes y = 3/4 x and y = − 3/4 x, and its closest distance to the center fountain
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a ⋅ b ⋅ π.A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.If the searchlight from the previous exercise has the light source located 6 inches from the base along the axis of symmetry and the opening is 4 feet, find the depth.
For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.The
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a ⋅ b ⋅ π.A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.An arch is in the shape of a parabola. It has a span of 100 feet and a maximum height of 20 feet. Find the equation of the parabola, and determine the height of the arch 40 feet from
For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a ⋅ b ⋅ π.A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus, and the other focus is 80 feet away, what is the length and
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.If the arch from the previous exercise has a span of 160 feet and a maximum height of 40 feet, find the equation of the parabola, and determine the distance from the center at which
For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.The
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.An object is projected so as to follow a parabolic path given by y = −x2 + 96x, where x is the horizontal distance traveled in feet and y is the height. Determine the maximum
For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.The
For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.The
For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.For the object from the previous exercise, assume the path followed is given by y = −0.5x2 + 80x. Determine how far along the horizontal the object traveled to reach maximum
For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix. r= 10 1 - 5 cos 0
For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.r(3 − 2sin θ) = 6
For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.16x2 + 64x − 4y2 − 8y − 4 = 0
For the following exercises, graph the parabola, labeling the focus and the directrix.x2 + 4x + 2y + 2 = 0
For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix. r = 6 3+2 cos 0
For the following exercises, graph the equation relative to the x′ y′ system in which the equation has no x′ y′ term.4x2 − 3√3 xy + y2 − 22 = 0
For the following exercises, graph the given ellipses, noting center, vertices, and foci.64x2 + 128x + 9y2 − 72y − 368 = 0
For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.r(6 − 4cos θ) = 5
For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.−100x2 + 1000x + y2 − 10y − 2575 = 0
For the following exercises, graph the parabola, labeling the focus and the directrix.y2 + 2y − 12x + 61 = 0
For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix. r = 1 4+3 sin 0
For the following exercises, graph the equation relative to the x′ y′ system in which the equation has no x′ y′ term.6x2 + 2√3 xy + 4y2 − 21 = 0
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: x = 4; e = 1/5
For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix. r= 3 55 sin 0
For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.4x2 + 16x − 4y2 + 16y + 16 = 0
For the following exercises, graph the parabola, labeling the focus and the directrix.−2x2 + 8x − 4y − 24 = 0
For the following exercises, graph the equation relative to the x′ y′ system in which the equation has no x′ y′ term.11x2 + 10√3 xy + y2 − 64 = 0
For the following exercises, graph the given ellipses, noting center, vertices, and foci.100x2 + 1000x + y2 − 10y + 2425 = 0
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.Directrix: x = − 4; e = 5
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.Directrix: y = 2; e = 2
For the following exercises, given information about the graph of the hyperbola, find its equation. Vertices at (3,0) and (−3,0) and one focus at (5,0).
For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci. r 3 1 sin 0
For the following exercises, graph the equation relative to the x′ y′ system in which the equation has no x′ y′ term.21x2 + 2√3 xy + 19y2 − 18 = 0
For the following exercises, graph the given ellipses, noting center, vertices, and foci.4x2 + 16x + 4y2 + 16y + 16 = 0
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.Directrix: y = − 2; e = 1/2
For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci. 8 4+3 sin 0
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