Chapter 7

Solve the system of equations. Give graphical support by making a sketch. $$\begin{aligned} &4 x^{2}+16 y^{2}=64\\\ &x^{2}+y^{2}=9 \end{aligned} $$

Write the given equation either in the form \((y-k)^{2}=a(x-h)\) or in the form \((x-h)^{2}=a(y-k)\). $$ y^{2}+8 x-8=4 x $$

Solve the system of equations. Give graphical support by making a sketch. $$\begin{aligned} 4 x^{2}+y^{2} &=4 \\ x^{2}+y^{2} &=2 \end{aligned} $$

Write the given equation either in the form \((y-k)^{2}=a(x-h)\) or in the form \((x-h)^{2}=a(y-k)\). $$ x=2 y^{2}+4 y-1 $$

Solve the system of equations. Give graphical support by making a sketch. $$ \begin{array}{r} x^{2}+y^{2}=9 \\ 2 x^{2}+3 y^{2}=18 \end{array} $$

Write the given equation either in the form \((y-k)^{2}=a(x-h)\) or in the form \((x-h)^{2}=a(y-k)\). $$ x=3 y^{2}-6 y-2 $$

Solve the system of equations. Give graphical support by making a sketch. $$ \begin{array}{r} x^{2}+y^{2}=4 \\ (x-1)^{2}+y^{2}=4 \end{array} $$

Write the given equation either in the form \((y-k)^{2}=a(x-h)\) or in the form \((x-h)^{2}=a(y-k)\). $$ x^{2}-3 x+4=2 y $$

Solve the system of equations. $$ \begin{aligned} \frac{x^{2}}{2}+\frac{y^{2}}{4} &=1 \\ -x^{2}+2 y &=4 \end{aligned} $$

Write the given equation either in the form \((y-k)^{2}=a(x-h)\) or in the form \((x-h)^{2}=a(y-k)\). $$ -3 y=-x^{2}+4 x-6 $$

Solve the system of equations. $$ \begin{aligned} &x^{2}+\frac{1}{9} y^{2}=1\\\ &x+y=3 \end{aligned} $$

Write the given equation either in the form \((y-k)^{2}=a(x-h)\) or in the form \((x-h)^{2}=a(y-k)\). $$ 4 y^{2}+4 y-5=5 x $$

Solve the system of equations. $$ \begin{aligned} &\frac{x^{2}}{2}+\frac{y^{2}}{4}=1\\\ &\frac{x^{2}}{4}+\frac{y^{2}}{2}=1 \end{aligned} $$

Write the given equation either in the form \((y-k)^{2}=a(x-h)\) or in the form \((x-h)^{2}=a(y-k)\). $$ -2 y^{2}+5 y+1=-x $$

Solve the system of equations. $$ \begin{aligned} &\frac{x^{2}}{5}+\frac{y^{2}}{10}=1\\\ &\frac{x^{2}}{10}+\frac{y^{2}}{5}=1 \end{aligned} $$

Graph the parabola. $$(y+0.75)^{2}=-3 x$$

Solve the system of equations. $$ \begin{aligned} (x-2)^{2}+y^{2} &=9 \\ x^{2}+y^{2} &=9 \end{aligned} $$

Graph the parabola. $$(y-3)^{2}=\frac{1}{7} x$$

Solve the system of equations. $$ \begin{aligned} (x-2)-y^{2} &=0 \\ \frac{x^{2}}{4}+\frac{y^{2}}{9} &=1 \end{aligned} $$

Graph the parabola. $$ (y-0.5)^{2}=3.1(x+1.3) $$

Shade the solutions set to the system. $$ \begin{aligned} (x-1)^{2}+(y+1)^{2} &<4 \\ (x+1)^{2}+y^{2} &>1 \end{aligned} $$

Graph the parabola. $$ 1.4(y-1.5)^{2}=0.5(x+2.1) $$

Shade the solutions set to the system. $$ \begin{aligned} &\frac{x^{2}}{16}+\frac{y^{2}}{25}<1\\\ &\frac{x^{2}}{4}+\frac{y^{2}}{9}>1 \end{aligned} $$

Graph the parabola. $$x=2.3(y+1)^{2}$$

Shade the solutions set to the system. $$ \begin{aligned} \frac{x^{2}}{4}+\frac{y^{2}}{9} & \leq 1 \\ x+y & \geq 2 \end{aligned} $$

Graph the parabola. $$(y-2.5)^{2}=4.1(x+1)$$

Shade the solutions set to the system. $$ \begin{aligned} &\frac{x^{2}}{16}+\frac{y^{2}}{25} \leq 1\\\ &-x+y \leq 4 \end{aligned} $$

Solve each system. $$ \begin{aligned} &x^{2}=2 y\\\ &x^{2}=y+1 \end{aligned} $$

Shade the solutions set to the system. $$ \begin{array}{r} x^{2}+y^{2} \leq 4 \\ x^{2}+(y-2)^{2} \leq 4 \end{array} $$

Solve each system. $$ \begin{aligned} x^{2} &=-3 y \\ -x^{2} &=2 y-2 \end{aligned} $$

Shade the solutions set to the system. $$ \begin{aligned} &x^{2}+(y+1)^{2} \leq 9\\\ &(x+1)^{2}+y^{2} \leq 9 \end{aligned} $$

Solve each system. $$ \begin{array}{l} y^{2}=-3 x \\ y^{2}=x+1 \end{array} $$

Shade the solutions set to the system. $$ \begin{aligned} x^{2}+y^{2} & \leq 4 \\ (x+1)^{2}-y & \leq 0 \end{aligned} $$

Solve each system. $$ \begin{aligned} -2 y^{2} &=x-5 \\ y^{2} &=2 x \end{aligned} $$

Shade the solutions set to the system. $$ \begin{aligned} 4 x^{2}+9 y^{2} & \leq 36 \\ x-(y-2)^{2} & \geq 0 \end{aligned} $$

Solve each system. $$ \begin{aligned} &(y-1)^{2}=x+1\\\ &(y+2)^{2}=-x+4 \end{aligned} $$

Shade the region in the xy-plane that satisfies the given inequality. Find the area of this region if units are in feet. $$ 4 x^{2}+9 y^{2} \leq 36 $$

Solve each system. $$\begin{array}{c} (y+1)^{2}=-x \\ -(y-1)^{2}=x+4 \end{array}$$

Shade the region in the xy-plane that satisfies the given inequality. Find the area of this region if units are in feet. $$ 9 x^{2}+y^{2} \leq 9 $$

Shade the region in the xy-plane that satisfies the given inequality. Find the area of this region if units are in feet. $$ \frac{(x-1)^{2}}{25}+\frac{(y+2)^{2}}{16} \leq 1 $$

Shade the region in the xy-plane that satisfies the given inequality. Find the area of this region if units are in feet. $$ \frac{(x+3)^{2}}{4}+\frac{(y-2)^{2}}{8} \leq 1 $$

Find an equation of the orbit for the planet. Graph its orbit and the location of the sun at a focus on the positive x-axis. $$ \text { Mercury: } e=0.206, a=0.387 $$

A comet sometimes travels along a parabolic path as it passes the sun. In this case the sun is located at the focus of the parabola and the comet passes the sun once, rather than orbiting the sun. Suppose the path of a comet is given by \(y^{2}=100 x,\) where units are in millions of miles. (a) Find the coordinates of the sun. (b) Find the minimum distance between the sun and the comet.

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 the equation of an ellipse that would satisfy this situation.

Halley's comet travels in an elliptical orbit with \(a=17.95\) and \(b=4.44\) and passes by Earth roughly every 76 years. Note that each unit represents one astronomical unit, or 93 million miles. The comet most recently passed by Earth in February 1986 (Source: M. Zeilik, Introductory Astronomy and Astrophysics.) (a) Write an equation for this orbit, centered at \((0,0)\) with major axis on the \(x\) -axis. (b) If the sun lies (at the focus) on the positive \(x\) -axis, approximate its coordinates. (c) Determine the maximum and minimum distances between Halley's comet and the sun.

Explain how the distance between the focus and the vertex of a parabola affects the shape of the parabola.

The perimeter of the Roman Colosseum is an ellipse with major axis 620 feet and minor axis 513 feet. Find the distance between the foci of this ellipse.

Explain how to determine the direction that a parabola opens, given the focus and the directrix.

Earth has a nearly circular orbit with \(e \approx 0.0167\) and \(a=93\) million miles. Approximate the minimum and maximum distances between Earth and the sun.

Perimeter of an Ellipse The perimeter \(P\) of an ellipse can be approximated by $$ P \approx 2 \pi \sqrt{\frac{a^{2}+b^{2}}{2}} $$ (a) Approximate the distance in miles that Mercury travels in one orbit of the sun if \(a=36.0, b=35.2\) and the units are in millions of miles. (b) If a planet has a circular orbit, does this formula give the exact perimeter? Explain.