Chapter 13

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ x^{2}+y^{2}+2 x+12 y-12=0 $$

Graph each system. $$ \left\\{\begin{array}{r} x^{2}+y^{2}>9 \\ y>x^{2} \end{array}\right. $$

Solve each nonlinear system of equations. $$ \left\\{\begin{array}{l} x^{2}+3 y^{2}=6 \\ x^{2}-3 y^{2}=10 \end{array}\right. $$

Graph each equation. $$ x^{2}-9 y^{2}=9 $$

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ x^{2}+y^{2}+6 x+10 y-2=0 $$

Graph each system. $$ \left\\{\begin{aligned} x^{2}+y^{2} & \leq 9 \\ y &

Solve each nonlinear system of equations. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=1 \\ y=x^{2}-9 \end{array}\right. $$

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ (x-7)^{2}+(y-2)^{2}=4 $$

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ (x+2)^{2}+(y-3)^{2}=7 $$

Graph each system. $$ \left\\{\begin{array}{l} \frac{x^{2}}{4}+\frac{y^{2}}{9} \geq 1 \\ x^{2}+y^{2} \geq 4 \end{array}\right. $$

Solve each nonlinear system of equations. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=36 \\ y=\frac{1}{6} x^{2}-6 \end{array}\right. $$

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ y=x^{2}+4 $$

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ (x+1)^{2}+(y-2)^{2}=5 $$

Graph each system. $$ \left\\{\begin{array}{l} x^{2}+(y-2)^{2} \geq 9 \\ \frac{x^{2}}{4}+\frac{y^{2}}{25}<1 \end{array}\right. $$

Solve each nonlinear system of equations. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=16 \\ y=-\frac{1}{4} x^{2}+4 \end{array}\right. $$

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ y=x^{2}+12 x+36 $$

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ x^{2}+y^{2}-4 x-8 y-2=0 $$

Graph each system. $$ \left\\{\begin{array}{r} x^{2}-y^{2} \geq 1 \\ y \geq 0 \end{array}\right. $$

Solve each nonlinear system of equations. $$ \left\\{\begin{array}{l} y=\sqrt{x} \\ x^{2}+y^{2}=12 \end{array}\right. $$

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ \frac{x^{2}}{4}+\frac{y^{2}}{9}=1 $$

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ x^{2}+y^{2}-2 x-6 y-5=0 $$

Graph each system. $$ \left\\{\begin{array}{r} x^{2}-y^{2} \geq 1 \\ x \geq 0 \end{array}\right. $$

Solve each nonlinear system of equations. $$ \left\\{\begin{array}{l} y=\sqrt{x} \\ x^{2}+y^{2}=20 \end{array}\right. $$

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ \frac{y^{2}}{9}-\frac{x^{2}}{9}=1 $$

Hint: For Exercises 33 through 38 , first divide the equation through by the coefficient of \(x^{2}\) (or \(\left.y^{2}\right)\). $$ 3 x^{2}+3 y^{2}=75 $$

Graph each system. $$ \left\\{\begin{array}{l} x+y \geq 1 \\ 2 x+3 y<1 \\ x>-3 \end{array}\right. $$

Graph each inequality in two variables. $$ x>-3 $$

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ \frac{x^{2}}{16}-\frac{y^{2}}{4}=1 $$

Hint: For Exercises 33 through 38 , first divide the equation through by the coefficient of \(x^{2}\) (or \(\left.y^{2}\right)\). $$ 2 x^{2}+2 y^{2}=18 $$

Graph each system. $$ \left\\{\begin{array}{l} x-y<-1 \\ 4 x-3 y>0 \\ y>0 \end{array}\right. $$

Graph each inequality in two variables. $$ y \leq 1 $$

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ \frac{x^{2}}{16}+\frac{y^{2}}{4}=1 $$

Hint: For Exercises 33 through 38 , first divide the equation through by the coefficient of \(x^{2}\) (or \(\left.y^{2}\right)\). $$ 6(x-4)^{2}+6(y-1)^{2}=24 $$

Graph each system. $$ \left\\{\begin{array}{l} x^{2}-y^{2}<1 \\ \frac{x^{2}}{16}+y^{2} \leq 1 \\ x \geq-2 \end{array}\right. $$

Graph each inequality in two variables. $$ y<2 x-1 $$

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ x^{2}+y^{2}=16 $$

Hint: For Exercises 33 through 38 , first divide the equation through by the coefficient of \(x^{2}\) (or \(\left.y^{2}\right)\). $$ 7(x-1)^{2}+7(y-3)^{2}=63 $$

Graph each system. $$ \left\\{\begin{array}{l} x^{2}-y^{2} \geq 1 \\ \frac{x^{2}}{16}+\frac{y^{2}}{4} \leq 1 \\ y \geq 1 \end{array}\right. $$

Graph each inequality in two variables. $$ 3 x-y \leq 4 $$

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ x=y^{2}+4 y-1 $$

Hint: For Exercises 33 through 38 , first divide the equation through by the coefficient of \(x^{2}\) (or \(\left.y^{2}\right)\). $$ 4(x+1)^{2}+4(y-3)^{2}=12 $$

Without graphing, how can you tell that the graphs of \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}=4\) do not have any points of intersection?

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ x=-y^{2}+6 y $$

Hint: For Exercises 33 through 38 , first divide the equation through by the coefficient of \(x^{2}\) (or \(\left.y^{2}\right)\). $$ 5(x-2)^{2}+5(y+1)=50 $$

Without solving, how can you tell that the graphs of \(y=2 x+3\) and \(y=2 x+7\) do not have any points of intersection?

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ 9 x^{2}-4 y^{2}=36 $$

Write an equation of the circle with the given center and radius. See Example 8. $$ (2,3) ; 6 $$

How many real solutions are possible for a system of equations whose graphs are a circle and a parabola?

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ 9 x^{2}+4 y^{2}=36 $$

Write an equation of the circle with the given center and radius. See Example 8. $$ (-7,6) ; 2 $$