Algebra and Trigonometry (4th Edition)

Chapter 14

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focus ; The focus is the point that is at the same... more

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; The value of is the distance from the vertex ... more

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; The focus of the parabola is where the focus ... more

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The equation is of the form that opens sidewards... more

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V ; The given equation of the parabola is .Since ... more

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IV ; The given equation of parabola is .Since the ... more

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I ; The equation of the parabola is 2x^{2}=y.... more

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VI ; The equation of the parabola is . ; Find the ... more

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Solve the given equation for .Divide both sides by... more

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Solve the given equation for .Divide both sides by... more

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Solve the given equation for .Determine the square... more

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Solve the given equation for .Divide both sides by... more

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Solve the given equation for .Subtract both sides ... more

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Solve the given equation for .Subtract both sides ... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Find the value of by using the given distance ... more

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; Find the value of by using the given distance ... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Write the equation of the directrix.Its -... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Find the value of .Since the parabola has ... more

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; Let be the focus of the parabola. Given that ... more

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; Let be the focus of the parabola. Substitute ... more

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25 ft ; Let the vertex of the parabola be the ... more

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; Let the origin be the vertex of the parabola. ... more

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660 in. ; Let the vertex of the parabola be the ... more

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Other scenarios wherein parabolas occur are in ... more

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Yes; Point tool F towards the floor such that the ... more

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sum ; An ellipse is a conic such that the sum of ... more

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; An ellipse with equation of the form such that ... more

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0 ; An ellipse with equation of the form such that... more

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The ellipse is of the form such that and . Since ... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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Solve the given equation for .Subtract both sides ... more

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Solve the given equation for .Subtract both sides ... more

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Solve the given equation for .Subtract both sides ... more

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Solve the given equation for .Subtract both sides ... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Find and given that the length of the major ... more

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; Find and given that the foci are at points... more

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; Find and given that the foci are at points... more

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; Find and given that the vertices are at ... more

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; Find and given that the endpoints of the ... more

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; Find and given that the length of the major ... more

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; Find and given that the length of the major ... more

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; Find and given that the foci are at points... more

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; Find and given that the foci are at points... more

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; Find and given that the length of the major ... more

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; Find and given that the length of the major ... more

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; Find the vertices and the endpoints of the ... more

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; Find the vertices and the endpoints of the ... more

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; Find the vertices and the endpoints of the ... more

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; Find the vertices and the endpoints of the ... more

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From the given equation, and . Using the ... more

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Label the endpoints of the left latus rectum. ; ... more

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; Draw an illustration of the ellipse to ... more

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; Find and given that the length of the minor ... more

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; Draw an illustration of the ellipse to ... more

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8 ft. ft. ; The maximum possible length of the ... more

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; Find and by referring to the given ... more

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Ask two classmates to hold each side of the string... more

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If the light beam is perpendicular to the wall, ... more

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Yes, it will look like an ellipse. ; The compass ... more

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difference ; The sum of the distances from the ... more

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horizontal ; The coefficient of is positive and ... more

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vertical ; The coefficient of is positive and is... more

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The coefficient of is positive and is negative. ... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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; Determine the form of the equation of the given... more

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Solve the given equation for .Subtract both sides ... more

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Solve the given equation for .Add both sides by .... more

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Add to both sides of the equation. ; Multiply both... more

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Subtract from both sides of the equation. ; ... more

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; Find the value of using the given vertices.... more

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; Find the value of using the given vertices.... more

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; Find the value of using the given vertices.... more

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; Find the value of using the given vertices.... more

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; Find the value of using the given vertices.... more

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; Find the value of using the given vertices.... more

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; Find the value of using the given vertices.... more

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; Find the value of using the given vertices.... more

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; Graph the given asymptotes and plot the given ... more

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; Graph the given asymptotes and plot the given ... more

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; Find the value of given the foci.Since the foci... more

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; Find the value of given the foci.Since the foci... more

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; Find the value of given the foci.Since the foci... more

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; Find the value of given the foci.Since the foci... more

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Adding to both sides, the equation becomes .... more

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; Since the transverse axis is along the x-axis, ... more

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Other scenarios wherein hyperbolas occur are those... more

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The boundary is a hyperbola since structure W ... more

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Focus: ; Replacing by shifts the graph 3 units to ... more

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Top EllipseVertex: Focus: Focus: Vertex: Bottom ... more

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Directrix: ; A hyperbola with equation of the form... more

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; The given parabola opens downward and has a ... more

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; The given parabola opens to the right and has a... more

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; The major axis of the given ellipse is ... more

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; Draw the major axis of the ellipse and label ... more

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; The hyperbola has a vertical transverse axis. ... more

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; Identify the vertices and the center of the ... more

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; Plot the given points and draw a sketch of the ... more

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; Plot the given points and draw a sketch of the ... more

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; Plot the given points and draw a sketch of the ... more

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; Plot the given points and draw a sketch of the ... more

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; Plot the given vertex and directrix. Draw a ... more

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; Plot the given vertex and directrix. Draw a ... more

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; Plot the given points and draw a sketch of the ... more

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; Plot the given points and draw a sketch of the ... more

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; Plot the given foci and the point. Draw a ... more

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; Plot the given foci and the point. Draw a ... more

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; Since the parabola has a horizontal axis of ... more

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; Since the parabola has a vertical axis of ... more

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Paraboladirectrix:   ; Expand the right-hand side ... more

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Ellipse major axis 6minor axis 4 ; Group and -... more

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Hyperbola;Center: ;Focus: ;Vertices: ;Asymptotes... more

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Parabola directrix: ; Isolate -term on the right-... more

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Ellipse; ;;, ;Length of major axis: 10;Length of ... more

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Ellipse; ;, ;;Length of major axis: Length of ... more

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Hyperbola; ;; ; asymptotes: ; Write the equation ... more

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Paraboladirectrix: ; Write the equation in the ... more

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Degenerate conic (pair of straight lines): ; ... more

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Hyperbola; ;;, ;Asymptotes: ; Complete the squares... more

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Point ; Complete the squares:Group the - and -... more

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No graph. The equation results to , which has no ... more

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Solve the equation for .  ;   ; Draw the graph ... more

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Since the equation is in the form , use the ... more

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Group the - and -terms.Since the equation is in ... more

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Group the - and the -terms.Since equation is in ... more

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; Isolate term.Factor out .  ;   ; Determine the... more

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; The vertex lies in the middle of the parabolic ... more

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; Write the length of the major axis as the sum ... more

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; The coordinates and of a point in the and ... more

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; The -coordinate of the new point after rotating... more

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; The -coordinate of the new point after rotating... more

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; The -coordinate of the new point after rotating... more

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; The given point corresponds to and .... more

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; The given point corresponds to and .... more

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; The given point corresponds to and .... more

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; Substitute into the rotation equations and... more

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; Substitute into the rotation equations and... more

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; Substitute into the rotation equations and .... more

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; Substitute into the rotation equations and .... more

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; Substitute into the rotation equations and... more

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; Substitute into the rotation equations and... more

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; Multiply the first equation by and the second ... more

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Hence, is part of a parabola by rotating the axes... more

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Since the distance between two points under ... more

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focus ; A conic is a set of all points P such that... more

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; A conic with one focus at the origin and ... more

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; Substitute for e and 3 for d into the equation... more

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; Substitute for e and 3 for d into the equation... more

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; Substitute 1 for e and 2 for d into the ... more

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; Substitute for e and 4 for d into the equation... more

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; Replace in with ; Multiply both sides of the ... more

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; Replace   in with ; Multiply both sides of the... more

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; Plug in 1 for e, 5 for r, and for   into   ... more

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; Use the formula for eccentricity   to find the ... more

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Since is of the form such that the equation is ... more

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III ; First, divide both numerator and denominator... more

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VI ; Since   is of the form   such that the ... more

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I ; First, divide both numerator and denominator ... more

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IV ; First, divide both numerator and denominator ... more

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V ; First, divide both numerator and denominator ... more

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When the value of is near to zero, the ellipse is... more

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; Write equations for the perihelion and aphelion... more

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That the focus of the conic is placed at the ... more

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Using polar coordinates and setting planet E as ... more

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; Since the graph is a parabola with vertex at ... more

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; Since the major axis of the ellipse is ... more

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; Since the graph is a parabola with a horizontal... more

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; Since the major axis of the ellipse is ... more

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; Since the graph is a hyperbola with horizontal ... more

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parabola ; Vertex: Focus: Directrix: ; Rewrite the... more

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ellipse ; Center: Foci: Vertices: , ; Rewrite the ... more

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hyperbola ; Center: Foci: Vertices: Asymptotes: ; ... more

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hyperbola ; Center: Foci: Vertices: and Asymptotes... more

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ellipse ; Center: Foci: Vertices: ; Rewrite the ... more

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parabola ; Vertex: Focus: Directrix: ; Simplify ... more

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parabola ; Vertex: Focus: Directrix: ; Add 64 to ... more

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hyperbola ; Center: Foci: Vertices: Asymptotes: ; ... more

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ellipse ; Center: Foci: Vertices: , ; Rewrite the ... more

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hyperbola ; Center: Foci: Vertices: and Asymptotes... more

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Has no graph ; Subtract 27 from both sides of the ... more

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ellipse ; Center: Foci: , Vertices: ; Rewrite the ... more

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; A parabola with vertex at the origin and a ... more

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; A parabola with vertex at the origin and a ... more

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; An ellipse centered at the origin with vertical... more

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; Use midpoint formula on the vertices of the ... more

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; If the foci of an ellipse are units away from ... more

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; If the foci of a hyperbola are units away from ... more

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; Use midpoint formula on the foci and of the ... more

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; Find the value of for the given parabola.Since ... more

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; Use midpoint formula on the vertices and of the... more

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; Use the vertex and intercept of parabola to ... more

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; Let points and on the axis be the foci of a ... more

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ellipse ; Write the equation for in terms of . ... more

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hyperbola ; Write the equation for in terms of .... more

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parabola ; Subtract from both sides of the ... more

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ellipse ; Subtract 100 from both sides and rewrite... more

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focus: , directrix: ; Find the value of for the ... more

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vertices: foci: length of major axis: 8length of ... more

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vertices: foci: asymptotes: ; Find the vertices of... more

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; Determine the value of .A parabola with ... more

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; Determine the value of . The coordinates of the... more

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; Determine the value of .The coordinates of the ... more

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; The given graph is a parabola that opens to the... more

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; The graph shown is an ellipse with a horizontal... more

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; Find the value of .Since the transverse axis of... more

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ellipse ; center: foci: vertices: and ; A conic of... more

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hyperbola ; center: foci: vertices: asymptotes: ; ... more

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parabola ; vertex: focus: directrix: ; A conic of ... more

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; The foci of an ellipse are units away from its ... more

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; Determine the value of .The focus of a parabola... more

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in ; Place the vertex of the parabola at the ... more

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parabola: Gardens by the Bay in Singaporeellipse: ... more

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Divide the length of the rectangle into equal ... more

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The more that the circular bases get twisted, the ... more

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