Chapter 8

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &\frac{x^{2}}{4}+\frac{y^{2}}{9} \leq 1\\\ &\frac{y^{2}}{4}-\frac{x^{2}}{9} \leq 1 \end{aligned}$$

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. $$y^{2}+8 y-3 x^{2}+13=0$$

Write an equation for each parabola with vertex at the origin. Through \((-3,3) ;\) opening to the left

Graph the solution set of each system of inequalities by hand. $$\begin{array}{c} 4 x^{2}-y^{2}>4 \\ 9 x^{2}+4 y^{2}>36 \end{array}$$

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. $$4 y^{2}+32 y-5 x^{2}-10 x+39=0$$

Write an equation for each parabola with vertex at the origin. Through \((2,-4)\); symmetric with respect to the \(y\) -axis

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &16 x^{2}+9 y^{2}<144\\\ &(x-1)^{2}+(y+1)^{2}>1 \end{aligned}$$

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. $$5 x^{2}+10 x-7 y^{2}+28 y=58$$

Write an equation for each parabola with vertex at the origin. Through \((3,2) ;\) symmetric with respect to the \(x\) -axis

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &\frac{(x-1)^{2}}{9}+\frac{y^{2}}{4} \leq 1\\\ &\frac{x^{2}}{4}-\frac{(y+1)^{2}}{9} \geq 1 \end{aligned}$$

RELATING CONCEPTS For individual or group investigation (Exercises \(85-90\) ) Consider the ellipse and hyperbola defined by $$ \frac{x^{2}}{16}+\frac{y^{2}}{12}=1 \quad \text { and } \quad \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 $$ respectively. Work Exercises \(85-90\) in order. Find the foci of the ellipse. Call them \(F_{1}\) and \(F_{2}\)

Find an equation of a parabola that satisfies the given conditions. Focus \((0,2) ;\) vertex \((0,1)\)

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &\frac{(x-2)^{2}}{36}+\frac{(y+2)^{2}}{25} \leq 1\\\ &\frac{(x+1)^{2}}{9}+\frac{(y-3)^{2}}{25} \leq 1 \end{aligned}$$

RELATING CONCEPTS For individual or group investigation (Exercises \(85-90\) ) Consider the ellipse and hyperbola defined by $$ \frac{x^{2}}{16}+\frac{y^{2}}{12}=1 \quad \text { and } \quad \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 $$ respectively. Work Exercises \(85-90\) in order. Graph the ellipse with a calculator, and trace to find the coordinates of several points on the ellipse.

Find an equation of a parabola that satisfies the given conditions. Focus \((-1,2) ;\) vertex \((3,2)\)

Solve each application. The orbit of Mars around the sun is an ellipse with equation $$ \frac{x^{2}}{5013}+\frac{y^{2}}{4970}=1 $$ where \(x\) and \(y\) are measured in millions of miles. Approximate the eccentricity \(e\) of this ellipse.

Find an equation of a parabola that satisfies the given conditions. Focus \((0,0) ;\) directrix \(x=-2\)

Solve each application. Neptune and Pluto both have elliptical orbits with the sun at one focus. Neptune's orbit has \(a=30.1\) astronomical units (AU) and eccentricity \(e=0.009,\) whereas Pluto's orbit has \(a=39.4 \mathrm{AU}\) and \(e=0.249 .(1 \mathrm{AU}\) is equal to the average distance from Earth to the sun and is approximately \(149,600,000\) kilometers.) (Source: Zeilik, M., S. Gregory, and E. Smith, Introductory Astronomy and Astrophysics, Saunders College Publishers.) (a) Position the sun at the origin, and determine an equation for each orbit. (b) Graph both equations on the same coordinate axes. Use the window \([-60,60]\) by \([-40,40]\)

Find an equation of a parabola that satisfies the given conditions. Focus \((2,1) ;\) directrix \(x=-1\)

Find an equation of a parabola that satisfies the given conditions. Focus \((-1,3) ;\) directrix \(y=7\)

Solve each application. The famous Halley's comet last passed Earth in February 1986 and will next return in 2062 . Halley's comet has an elliptical orbit of eccentricity 0.9673 with the sun at one of the foci. The greatest distance of the comet from the sun is 3281 million miles. Find the least distance between Halley's comet and the sun. (Source: The World Almanac and Book of Facts.)

Find an equation of a parabola that satisfies the given conditions. Focus \((1,2) ;\) directrix \(y=4\)

Solve each application. The orbit of Earth is an ellipse with the sun at one focus. The distance between Earth and the sun ranges from 91.4 to 94.6 million miles. Estimate the eccentricity of Earth's orbit. (Source: The World Almanac and Book of Facts.)

Find an equation of a parabola that satisfies the given conditions. Horizontal axis; vertex \((-2,3)\); passing through \((-4,0)\)

Solve each problem. Orbit of Venus The orbit of Venus is an ellipse, with the sun at one focus. An approximate equation for the orbit is $$ \frac{x^{2}}{5013}+\frac{y^{2}}{4970}=1 $$ where \(x\) and \(y\) are measured in millions of miles. (a) Approximate the length of the major axis. (b) Approximate the length of the minor axis.

Find an equation of a parabola that satisfies the given conditions. Horizontal axis; vertex \((-1,2) ;\) passing through \((2,3)\)

Solve each problem. The Roman Colosseum The Roman Colosseum is an ellipse with major axis 620 feet and minor axis 513 feet. Approximate the distance between the foci of this ellipse.

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=(x+3)^{2}-4$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=(x-5)^{2}-4$$

Height of an Overpass A one-way road passes under an overpass in the form of half of an ellipse 15 feet high at the center and 20 feet wide. Assuming that a truck is 12 feet wide, what is the height of the tallest truck that can pass under the overpass? (GRAPH CANT COPY)

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=-2(x+3)^{2}+2$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=\frac{2}{3}(x-2)^{2}-1$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=x^{2}-2 x+3$$

Sound Detection Microphones are placed at points \((-c, 0)\) and \((c, 0) .\) An explosion occurs at point \(P(x, y)\) having positive \(x\) -coordinate. The sound is detected at the closer microphone \(t\) seconds before being detected at the farther microphone. Assume that sound travels at a speed of 330 meters per second, and show that \(P\) must be on the hyperbola $$ \frac{x^{2}}{330^{2} t^{2}}-\frac{y^{2}}{4 c^{2}-330^{2} t^{2}}=\frac{1}{4} $$ (GRAPH CANT COPY)

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=x^{2}+6 x+5$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=2 x^{2}-4 x+5$$

Equation of a Hyperbola Suppose a hyperbola has center at the origin, foci at \(F^{\prime}(-c, 0)\) and \(F(c, 0),\) and equation $$ \left|d\left(P, F^{\prime}\right)-d(P, F)\right|=2 a $$ Let \(b^{2}=c^{2}-a^{2},\) and show that the points on the hyperbola satisfy the equation $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 $$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=-3 x^{2}+24 x-46$$

Equation of an Ellipse Use the definition of an ellipse to find an equation of an ellipse with foci \((3,0)\) and \((-3,0),\) where the sum of the distances from any point of the ellipse to the two foci is \(10 .\)

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=y^{2}+2$$

Equation of a Hyperbola Use the definition of a hyperbola to find an equation of a hyperbola with center at the origin, foci \((-2,0)\) and \((2,0),\) and the absolute value of the difference of the distances from any point of the hyperbola to the two foci equal to 2 .

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=(y+1)^{2}$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=(y-3)^{2}$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$(y+2)^{2}=x+1$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=(y-4)^{2}+2$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=-2(y+3)^{2}$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=\frac{2}{3} y^{2}-4 y+8$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=y^{2}+2 y-8$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=-4 y^{2}-4 y-3$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=-2 y^{2}+2 y-3$$