Section 1
The Circle
In Exercises $1-10,$ write the standard form of the equation of the circle with the given center and radius.$$\text { Center }(0,0), r=7$$
In Exercises $1-10,$ write the standard form of the equation of the circle with the given center and radius.$$\text { Center }(0,0), r=8$$
In Exercises $1-10,$ write the standard form of the equation of the circle with the given center and radius.$$\text { Center }(3,2), r=5$$
In Exercises $1-10,$ write the standard form of the equation of the circle with the given center and radius.$$\text { Center }(2,-1), r=4$$
In Exercises $1-10,$ write the standard form of the equation of the circle with the given center and radius.$$\text { Center }(-1,4), r=2$$
In Exercises $1-10,$ write the standard form of the equation of the circle with the given center and radius.$$\text { Center }(-3,5), r=3$$
In Exercises $1-10,$ write the standard form of the equation of the circle with the given center and radius.$$\text { Center }(-3,-1), r=\sqrt{3}$$
In Exercises $1-10,$ write the standard form of the equation of the circle with the given center and radius.$$\text { Center }(-5,-3), r=\sqrt{5}$$
In Exercises $1-10,$ write the standard form of the equation of the circle with the given center and radius.$$\text { Center }(-4,0), r=10$$
In Exercises $1-10,$ write the standard form of the equation of the circle with the given center and radius.$$\text { Center }(-2,0), r=6$$
In Exercises $11-18,$ give the center and radius of the circle described by the equation and graph each equation.$$x^{2}+y^{2}=16$$
In Exercises $11-18,$ give the center and radius of the circle described by the equation and graph each equation.$$x^{2}+y^{2}=49$$
In Exercises $11-18,$ give the center and radius of the circle described by the equation and graph each equation.$$(x-3)^{2}+(y-1)^{2}=36$$
In Exercises $11-18,$ give the center and radius of the circle described by the equation and graph each equation.$$(x-2)^{2}+(y-3)^{2}=16$$
In Exercises $11-18,$ give the center and radius of the circle described by the equation and graph each equation.$$(x+3)^{2}+(y-2)^{2}=4$$
In Exercises $11-18,$ give the center and radius of the circle described by the equation and graph each equation.$$(x+1)^{2}+(y-4)^{2}=25$$
In Exercises $11-18,$ give the center and radius of the circle described by the equation and graph each equation.$$(x+2)^{2}+(y+2)^{2}=4$$
In Exercises $11-18,$ give the center and radius of the circle described by the equation and graph each equation.$$(x+4)^{2}+(y+5)^{2}=36$$
In Exercises $19-26,$ complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.$$x^{2}+y^{2}+6 x+2 y+6=0$$
In Exercises $19-26,$ complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.$$x^{2}+y^{2}+8 x+4 y+16=0$$
In Exercises $19-26,$ complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.$$x^{2}+y^{2}-10 x-6 y-30=0$$
In Exercises $19-26,$ complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.$$x^{2}+y^{2}-4 x-12 y-9=0$$
In Exercises $19-26,$ complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.$$x^{2}+y^{2}+8 x-2 y-8=0$$
In Exercises $19-26,$ complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.$$x^{2}+y^{2}+12 x-6 y-4=0$$
In Exercises $19-26,$ complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.$$x^{2}-2 x+y^{2}-15=0$$
In Exercises $19-26,$ complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.$$x^{2}+y^{2}-6 y-7=0$$
In Exercises $27-30,$ find the solution set for each system by graphing both of the system's equations in the same rectangularcoordinate system and finding all points of intersection. Check all solutions in both equations.$$\left\{\begin{array}{c}x^{2}+y^{2}=16 \\x-y=4\end{array}\right.$$
In Exercises $27-30,$ find the solution set for each system by graphing both of the system's equations in the same rectangularcoordinate system and finding all points of intersection. Check all solutions in both equations.$$\left\{\begin{array}{r}x^{2}+y^{2}=9 \\x-y=3\end{array}\right.$$
In Exercises $27-30,$ find the solution set for each system by graphing both of the system's equations in the same rectangularcoordinate system and finding all points of intersection. Check all solutions in both equations.$$\left\{\begin{aligned}(x-2)^{2}+(y+3)^{2} &=4 \\y &=x-3\end{aligned}\right.$$
In Exercises $27-30,$ find the solution set for each system by graphing both of the system's equations in the same rectangularcoordinate system and finding all points of intersection. Check all solutions in both equations.$$\left\{\begin{aligned}(x-3)^{2}+(y+1)^{2} &=9 \\y &=x-1\end{aligned}\right.$$
In Exercises $31-34$, write the standard form of the equation of the circle with the given graph.(GRAPH CAN'T COPY)
In Exercises $35-36,$ a line segment through the center of each circle intersects the circle at the points shown.a. Find the coordinates of the circle's center.b. Find the radius of the circle.c. Use your answers from parts (a) and (b) to write thestandard form of the circle's equation.(GRAPH CAN'T COPY)
A rectangular coordinate system with coordinates in miles is placed with the origin at the center of Los Angeles. The figure indicates that the University of Southern California is located 2.4 miles west and 2.7 miles south of central Los Angeles. A seismograph on the campus shows that a small earthquake occurred. The quake's epicenter is estimated to be approximately 30 miles from the university. Write the standard form of the equation for the set of points that could be the epicenter of the quake.(IMAGE CAN'T COPY)
The Ferris wheel in the figure has a radius of 68 feet. The clearance between the wheel and the ground is 14 feet. The rectangular coordinate system shown has its origin on the ground directly below the center of the wheel. Use the coordinate system to write the equation of the circular wheel.(IMAGE CAN'T COPY)
What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.
Give an example of a circle's equation in standard form. Describe how to find the center and radius for this circle.
How is the standard form of a circle's equation obtained from its general form?
Does $(x-3)^{2}+(y-5)^{2}=0$ represent the equation of a circle? If not, describe the graph of this equation.
Does $(x-3)^{2}+(y-5)^{2}=-25$ represent the equation of a circle? What sort of set is the graph of this equation?
In Exercises $44-46,$ use a graphing utility to graph each circle whose equation is given.$$x^{2}+y^{2}=25$$
In Exercises $44-46,$ use a graphing utility to graph each circle whose equation is given.$$(y+1)^{2}=36-(x-3)^{2}$$
In Exercises $44-46,$ use a graphing utility to graph each circle whose equation is given.$$x^{2}+10 x+y^{2}-4 y-20=0$$
In Exercises $47-50$, determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.I graphed $x^{2}+y^{2}=0$ as a circle with center $(0,0)$ and radius 0
In Exercises $47-50$, determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.To avoid sign errors when finding $h$ and $k,$ I place parentheses around the numbers that follow the subtraction signs in a circle's equation.
In Exercises $47-50$, determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.I used the equation $(x+1)^{2}+(y-5)^{2}=-4$ to identify the circle's center and radius.
In Exercises $47-50$, determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.My graph of $(x-2)^{2}+(y+1)^{2}=16$ is my graph of $x^{2}+y^{2}=16$ translated two units right and one unit down.
In Exercises $51-54,$ determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.The equation of the circle whose center is at the origin with radius 16 is $x^{2}+y^{2}=16$
In Exercises $51-54,$ determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.The graph of $(x-3)^{2}+(y+5)^{2}=36$ is a circle with radius 6 centered at $(-3,5)$
In Exercises $51-54,$ determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.The graph of $(x-4)+(y+6)=25$ is a circle with radius 5 centered at $(4,-6)$
In Exercises $51-54,$ determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.The graph of $(x-3)^{2}+(y+5)^{2}=-36$ is a circle with radius 6 centered at $(3,-5)$
Find the area of the doughnut-shaped region bounded by the graphs of $(x-2)^{2}+(y+3)^{2}=25$ and $(x-2)^{2}+(y+3)^{2}=36$
A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write the point-slope form of the equation of a line tangent to the circle whose equation is $x^{2}+y^{2}=25$ at the point$(3,-4)$
If $f(x)=x^{2}-2$ and $g(x)=3 x+4,$ find $f(g(x))$ and $g(f(x)) .$ (Section 8.4, Example 1)
Solve: $2 x=\sqrt{7 x-3}+3 .$ (Section 10.6, Example 3)
Solve: $|2 x-5|<10 .$ (Section 9.3, Example 4)
Exercises $60-62$ will help you prepare for the material covered in the next section.Set $y=0$ and find the $x$ -intercepts: $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$
Exercises $60-62$ will help you prepare for the material covered in the next section.Set $x=0$ and find the $y$ -intercepts: $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$
Exercises $60-62$ will help you prepare for the material covered in the next section.Divide both sides of $25 x^{2}+16 y^{2}=400$ by 400 andsimplify.