Chapter 10

Write an equation for the circle that satisfies each set of conditions. center \((-8,7),\) radius \(\frac{1}{2}\) unit

Find the distance between each pair of points with the given coordinates. $$ (1,-14),(-6,10) $$

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{y^{2}=x^{2}-25} \\ {x^{2}-y^{2}=7}\end{array} $$

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. $$ \frac{y^{2}}{16}-\frac{x^{2}}{25}=1 $$

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ x^{2}-y^{2}+8 x=16 $$

Graph each equation. $$ x=y^{2}-14 y+25 $$

Write an equation for the circle that satisfies each set of conditions. endpoints of a diameter at \((-5,2)\) and \((3,6)\)

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. \(\frac{y^{2}}{10}+\frac{x^{2}}{5}=1\)

Find the distance between each pair of points with the given coordinates. $$ (-4,-10),(-3,-11) $$

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{y^{2}=x^{2}-7} \\ {x^{2}+y^{2}=25}\end{array} $$

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. $$ \frac{x^{2}}{9}-\frac{y^{2}}{25}=1 $$

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ x^{2}+y^{2}-8 x-6 y+5=0 $$

Graph each equation. $$ x=5 y^{2}+25 y+60 $$

Write an equation for the circle that satisfies each set of conditions. endpoints of a diameter at \((11,18)\) and \((-13,-19)\)

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. $$ \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 $$

Find the distance between each pair of points with the given coordinates. $$ (9,-2),(12,-14) $$

Solve each system of inequalities by graphing. $$ \begin{array}{l}{x+2 y>1} \\ {x^{2}+y^{2} \leq 25}\end{array} $$

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. $$ \frac{(y-4)^{2}}{16}-\frac{(x+2)^{2}}{9}=1 $$

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ 3 x^{2}-2 y^{2}+32 y-134=0 $$

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

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. $$ \frac{(x+8)^{2}}{144}+\frac{(y-2)^{2}}{81}=1 $$

Find the distance between each pair of points with the given coordinates. $$ (0.23,0.4),(0.68,-0.2) $$

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

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. $$ \frac{(y-3)^{2}}{25}-\frac{(x-2)^{2}}{16}=1 $$

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ y^{2}+18 y-2 x=-84 $$

MANUFACTURING The reflective surface in a flashlight has a parabolic shape with a cross section that can be modeled by \(y=\frac{1}{3} x^{2},\) where \(x\) and \(y\) are in centimeters. How far from the vertex should the filament of the light bulb be located?

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+y^{2}=144 $$

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. $$ \frac{(y+11)^{2}}{144}+\frac{(x-5)^{2}}{121}=1 $$

Find the distance between each pair of points with the given coordinates. $$ (2.3,-1.2),(-4.5,3.7) $$

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

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

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ 7 x^{2}-28 x+4 y^{2}+8 y=-4 $$

Find the center and radius of the circle with the given equation. Then graph the circle. $$ (x-3)^{2}+(y-1)^{2}=25 $$

The rounded top of the window is the top half of an ellipse. Write an equation for the ellipse if the origin is at the midpoint of the bottom edge of the window. $$ 3 x^{2}+9 y^{2}=27 $$

GEOMETRY Quadrilateral RSTV has vertices \(R(-4,6), S(4,5), T(6,3),\) and \(V(5,-8) .\) Find the perimeter of the quadrilateral.

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

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. $$ x^{2}-y^{2}=4 $$

FOOTBALL When a ball is thrown or kicked, the path it travels is shaped like a parabola. Suppose a football is kicked from ground level, reaches a maximum height of 25 feet, and hits the ground 100 feet from where it was kicked. Assuming that the ball was kicked at the origin, write an equation of the parabola that models the flight of the ball.

Find the center and radius of the circle with the given equation. Then graph the circle. $$ (x+3)^{2}+(y+7)^{2}=81 $$

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 \(27 x^{2}+9 y^{2}=81\)

Solve each system of inequalities by graphing. $$ \begin{array}{l}{y^{2}

For Exercises \(23-25,\) match each equation below with the situation that it could represent. a. \(9 x^{2}+4 y^{2}-36=0\) b. \(0.004 x^{2}-x+y-3=0\) c. \(x^{2}+y^{2}-20 x+30 y-75=0\) PHOTOGRAPHY the oval opening in a picture frame

For Exercises \(24-27,\) use the equation \(x=3 y^{2}+4 y+1\) Draw the graph. Find the \(x\) -intercept \((\mathrm{s})\) and \(y\) -intercept \((\mathrm{s})\)

Find the center and radius of the circle with the given equation. Then graph the circle. $$ (x-3)^{2}+y^{2}=16 $$

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. \(7 x^{2}+3 y^{2}-28 x-12 y=-19\)

Find the midpoint of the line segment with endpoints at the given coordinates. Then find the distance between the points. $$ \left(-3,-\frac{2}{11}\right),\left(5, \frac{9}{11}\right) $$

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

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. $$ 6 y^{2}=2 x^{2}+12 $$

For Exercises \(24-27,\) use the equation \(x=3 y^{2}+4 y+1\) What is the equation of the axis of symmetry?

Find the center and radius of the circle with the given equation. Then graph the circle. $$ (x-3)^{2}+(y+7)^{2}=50 $$