Stewart Precalc 6e Section 11.4: Shifted Conics

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Stewart Precalc 6e Section 11.4: Shifted Conics 0/10
1 Skills - Complete the square · Level 3
Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. Then sketch the graph. \(x^2 - 4y^2 + 4x + 8y = 0\)
2 Problem - Conic from constant · Level 3
Determine what the value of \(F\) must be if the graph of the equation \(4x^2 + y^2 + 4(x - 2y) + F = 0\) is (a) an ellipse, (b) a single point, or (c) the empty set.
3 Problem - Composite conic · Level 3
Find an equation for the ellipse that shares a vertex and a focus with the parabola \(x^2 + y = 100\) and has its other focus at the origin.
4 Problem - Confocal parabolas · Level 4
This exercise deals with confocal parabolas, that is, families of parabolas that have the same focus.
(a) Draw graphs of the family of parabolas \(x^2 = 4p(y + p)\) for \(p = -2, -\dfrac{3}{2}, -1, -\dfrac{1}{2}, \dfrac{1}{2}, 1, \dfrac{3}{2}, 2\).
(b) Show that each parabola in this family has its focus at the origin.
(c) Describe the effect on the graph of moving the vertex closer to the origin.

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5 Application - Cannonball · Level 3
Path of a Cannonball. A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 ft from the cannon and the highest point it reaches is 3200 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.
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6 Application - Satellite orbit · Level 3
Orbit of a Satellite. A satellite is in an elliptical orbit around the earth with the center of the earth at one focus. The height of the satellite above the earth varies between 140 mi and 440 mi. Assume that the earth is a sphere with radius 3960 mi. Find an equation for the path of the satellite with the origin at the center of the earth.
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7 Discovery - Confocal conics · Level 4
A Family of Confocal Conics. Conics that share a focus are called confocal. Consider the family of conics that have a focus at \((0, 1)\) and a vertex at the origin, as shown in the figure.
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(a) Find equations of two different ellipses that have these properties.
(b) Find equations of two different hyperbolas that have these properties.
(c) Explain why only one parabola satisfies these properties. Find its equation.
(d) Sketch the conics you found in parts (a), (b), and (c) on the same coordinate axes (for the hyperbolas, sketch the top branches only).
(e) How are the ellipses and hyperbolas related to the parabola?

Enter your answer directly below each part above.

8 Example - Shifted ellipse · Level 2
Sketch the graph of the ellipse \(\dfrac{(x + 1)^2}{4} + \dfrac{(y - 2)^2}{9} = 1\) and determine the coordinates of the foci.
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9 Example - Shifted parabola · Level 2
Determine the vertex, focus, and directrix, and sketch a graph of the parabola \(x^2 - 4 x = 8 y - 28\).
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10 Example - Shifted hyperbola · Level 3
A shifted conic has the equation \(9 x^2 - 72 x - 16 y^2 - 32 y = 16\). (a) Complete the square in \(x\) and \(y\) to show that the equation represents a hyperbola. (b) Find the center, vertices, foci, and asymptotes of the hyperbola, and sketch its graph. (c) Draw the graph using a graphing calculator.

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