Chapter 10

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{y=5} \\ {y^{2}=x^{2}+9}\end{array} $$

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ y=x^{2}+3 x+1 $$

Write \(y=2 x^{2}-12 x+6\) in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola.

Find the midpoint of the line segment with endpoints at the given coordinates. $$ (-5,6),(1,7) $$

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{y-x=1} \\ {x^{2}+y^{2}=25}\end{array} $$

A hyperbola has foci at \((4,0)\) and \((-4,0) .\) The value of \(a\) is \(1 .\) Write an equation for the hyperbola.

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ y^{2}-2 x^{2}-16=0 $$

For Exercises 2 and \(3,\) use the following information. In order for a satellite to remain in a circular orbit above the same spot on Earth, the satellite must be 35,800 kilometers above the equator. 2\. Write an equation for the orbit of the satellite. Use the center of Earth as the origin and 6400 kilometers for the radius of Earth.

Graph each equation. $$ y=(x-3)^{2}-4 $$

Write an equation for the ellipse that satisfies each set of conditions. endpoints of major axis at (2, 2) and (2, -10), endpoints of minor axis at (0, -4) and (4, -4)

Find the midpoint of the line segment with endpoints at the given coordinates. $$ (8,9),(-3,-4.5) $$

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{3 x=8 y^{2}} \\ {8 y^{2}-2 x^{2}=16}\end{array} $$

Comets or other objects that pass by Earth or the Sun only once and never return may follow hyperbolic paths. Suppose a comet's path can be modeled by a branch of the hyperbola with equation \(\frac{y^{2}}{225}-\) \(\frac{x^{2}}{400}=1 .\) Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola. Then graph the hyperbola.

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ x^{2}+y^{2}=x+2 $$

Graph each equation. $$ y=2(x+7)^{2}+3 $$

Write an equation for the ellipse that satisfies each set of conditions. Write an equation for the ellipse that satisfies each set of conditions. endpoints of major axis at (0,10) and \((0,-10),\) foci at (0,8) and (0,-8)

Find the midpoint of the line segment with endpoints at the given coordinates. $$ (13,-4),(10,14.6) $$

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{5 x^{2}+y^{2}=30} \\ {9 x^{2}-y^{2}=-16}\end{array} $$

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ x^{2}+4 y^{2}+2 x-24 y+33=0 $$

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. $$ \frac{y^{2}}{18}-\frac{x^{2}}{20}=1 $$

Write an equation for the circle that satisfies each set of conditions. center \((-1,-5),\) radius 2 units

Graph each equation. $$ y=-3 x^{2}-8 x-6 $$

Write an equation for the ellipse that satisfies each set of conditions. At its closest point, Earth is 0.99 astronomical units from the center of the Sun. At its farthest point, Earth is 1.021 astronomical units from the center of the Sun. Write an equation for the orbit of Earth, assuming that the center of the orbit is the origin and the Sun lies on the \(x\)-axis.

Find the midpoint of the line segment with endpoints at the given coordinates. $$ (-12,-2),(-3.5,-7) $$

CELL PHONES A person using a cell phone can be located in respect to three cellular towers. In a coordinate system where a unit represents one mile, the caller is determined to be 50 miles from a tower at the origin, 40 miles from a tower at \((0,30),\) and 13 miles from a tower at \((35,18) .\) Where is the caller?

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. $$ \frac{(y+6)^{2}}{20}-\frac{(x-1)^{2}}{25}=1 $$

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ y^{2}-x-10 y+34=0 $$

Write an equation for the circle that satisfies each set of conditions. endpoints of a diameter at \((-4,1)\) and \((4,-5)\)

Graph each equation. $$ x=\frac{2}{3} y^{2}-6 y+12 $$

Find the coordinates of the center and foci and the lengths of the major and minor axes for the ellipse with the given equation. Then graph the ellipse. \(\frac{y^{2}}{18}+\frac{x^{2}}{9}=1\)

Find the distance between each pair of points with the given coordinates. $$ (2,-4),(10,-10) $$

Solve each system of inequalities by graphing. $$ \begin{array}{l}{x+y<4} \\ {9 x^{2}-4 y^{2} \geq 36}\end{array} $$

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. $$ x^{2}-36 y^{2}=36 $$

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ 3 x^{2}+2 y^{2}+12 x-28 y+104=0 $$

Write an equation for the circle that satisfies each set of conditions. endpoints of a diameter at \((2,-2)\) and \((-2,-6)\)

COMMUNICATION A microphone is placed at the focus of a parabolic reflector to collect sound for the television broadcast of a football game. Write an equation for the cross section, assuming that the focus is at the origin, the focus is 6 inches from the vertex, and the parabola opens to the right.

Find the coordinates of the center and foci and the lengths of the major and minor axes for the ellipse with the given equation. Then graph the ellipse. \(\frac{(x-1)^{2}}{20}+\frac{(y+2)^{2}}{4}=1\)

Find the distance between each pair of points with the given coordinates. $$ (7,8),(-4,9) $$

Solve each system of inequalities by graphing. $$ \begin{array}{l}{x^{2}+y^{2}<25} \\ {4 x^{2}-9 y^{2}<36}\end{array} $$

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. $$ 5 x^{2}-4 y^{2}-40 x-16 y-36=0 $$

When an airplane flies faster than the speed of sound, it produces a shock wave in the shape of a cone. Suppose the shock wave generated by a jet intersects the ground in a curve that can be modeled by the equation \(x^{2}-14 x+4=9 y^{2}-36 y .\) Identify the shape of the curve.

Find the center and radius of the circle with the given equation. Then graph the circle. $$ (x-4)^{2}+(y-1)^{2}=9 $$

Write each equation in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola. $$ y=x^{2}-6 x+11 $$

Find the coordinates of the center and foci and the lengths of the major and minor axes for the ellipse with the given equation. Then graph the ellipse. \(4 x^{2}+8 y^{2}=32\)

Find the distance between each pair of points with the given coordinates. $$ (0,5,1,4),(1,1,2,9) $$

Find the exact solution(s) of each system of equations. $$\begin{array}{l}{y=x+2} \\ {y=x^{2}}\end{array}$$

When an airplane flies faster than the speed of sound, it produces a shock wave in the shape of a cone. Suppose the shock wave generated by a jet intersects the ground in a curve that can be modeled by the equation \(x^{2}-14 x+4=9 y^{2}-36 y .\) Graph the equation.

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+(y-14)^{2}=34 $$

Write each equation in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola. $$ x=y^{2}+14 y+20 $$

Find the coordinates of the center and foci and the lengths of the major and minor axes for the ellipse with the given equation. Then graph the ellipse. \(x^{2}+25 y^{2}-8 x+100 y+91=0\)