Chapter 7

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

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. \(y^{2}+4 y-x^{2}+2 x=6\)

Write an equation for each parabola with vertex at the origin. $$\text { Focus }\left(0, \frac{1}{4}\right)$$

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &x^{2}+3 x y+y^{2}=5\\\ &x^{2}-2 x y-y^{2}=-7 \end{aligned}$$

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. \(3 y^{2}+24 y-2 x^{2}+12 x+24=0\)

Write an equation for each parabola with vertex at the origin. Through \((2,-2 \sqrt{2}) ;\) opening to the right

Use substitution to solve the nonlinear system of equations in three variables. Note that solutions are ordered triples. $$\begin{aligned} 2 x^{2}+y^{2}+3 z^{2} &=3 \\ 2 x+y-z &=1 \\ x+y &=0 \end{aligned}$$

Write an equation for each parabola with vertex at the origin. Through \((\sqrt{3}, 3) ;\) opening upward

Use substitution to solve the nonlinear system of equations in three variables. Note that solutions are ordered triples. $$\begin{aligned} x^{2}+y^{2}+z^{2} &=4 \\ x+y+z &=2 \\ x-y &=0 \end{aligned}$$

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. \(4 x^{2}+16 x-9 y^{2}+18 y=29\)

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. \(x^{2}-6 x-2 y^{2}+7=0\)

Write an equation for each parabola with vertex at the origin. Through \((\sqrt{10},-5) ;\) opening downward

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &\frac{x^{2}}{16}+\frac{y^{2}}{4} \leq 1\\\ &x^{2}-y^{2} \geq 1 \end{aligned}$$

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. $$\begin{aligned} &\frac{x^{2}}{4}+\frac{y^{2}}{9} \leq 1\\\ &\frac{y^{2}}{4}-\frac{x^{2}}{9} \leq 1 \end{aligned}$$

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

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. $$\begin{array}{c} 4 x^{2}-y^{2}>4 \\ 9 x^{2}+4 y^{2}>36 \end{array}$$

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

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. $$\begin{aligned} &16 x^{2}+9 y^{2}<144\\\ &(x-1)^{2}+(y+1)^{2}>1 \end{aligned}$$

For individual or group investigation. 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. 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. $$\begin{aligned} &\frac{(x-1)^{2}}{9}+\frac{y^{2}}{4} \leq 1\\\ &\frac{x^{2}}{4}-\frac{(y+1)^{2}}{9} \geq 1 \end{aligned}$$

For individual or group investigation. 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. Graph the ellipse with your 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 \((3,2)\)

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

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

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 \((2,1) ;\) directrix \(x=-1\)

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 \(a=39.4\) 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 \((-1,3) ;\) directrix \(y=7\)

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

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. (Source: The World Almanac and Book of Facts. Find the least distance between Halley's comet and the sun.

Solve each problem. A patient's kidney stone is placed 12 units away from the source of the shock waves of a lithotripter. The lithotripter is based on an ellipse with a minor axis that measures 16 units. Find an equation of an ellipse that would satisfy this situation.

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

Solve each application. Orbit of Earth 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. (Source: The World Almanac and Book of Facts.) Estimate the eccentricity of Earth's orbit.

Solve each problem. 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 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. $$y=(x-5)^{2}-4$$

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. $$y=x^{2}-2 x+3$$

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

Solve each problem. Structure of an Atom In \(1911,\) Ernest Rutherford discovered the basic structure of the atom by "shooting" positively charged alpha particles with a speed of \(10^{7}\) meters per second at a piece of gold foil \(6 \times 10^{-7}\) meter thick. Only a small percentage of the alpha particles struck a gold nucleus head-on and were deflected directly back toward their source. The rest of the particles often followed a hyperbolic trajectory because they were repelled by positively charged gold nuclei. Thus, Rutherford proposed that the atom was composed of mostly empty space and a small, dense nucleus. The figure shows an alpha particle \(A\) initially approaching a gold nucleus \(N\) and being deflected at an angle \(\theta=90^{\circ}\) \(N\) is located at a focus of the hyperbola, and the trajectory of \(A\) passes through a vertex of the hyperbola. (a) Determine the equation of the trajectory of the alpha particle if \(d=5 \times 10^{-14}\) meter. (b) Approximate the minimum distance between the centers of the alpha particle and the gold nucleus. (GRAPH CAN'T COPY)

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

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 $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$