### Review Exercises

##### Distance and Midpoint Formulas; Circles

**Use the Distance Formula**

In the following exercises, find the distance between the points. Round to the nearest tenth if needed.

$\left(\mathrm{-5},1\right)$ and $\left(\mathrm{-1},4\right)$

$\left(8,2\right)$ and $\left(\mathrm{-7},\mathrm{-3}\right)$

**Use the Midpoint Formula**

In the following exercises, find the midpoint of the line segments whose endpoints are given.

$\left(\mathrm{-2},\mathrm{-6}\right)$ and $\left(\mathrm{-4},\mathrm{-2}\right)$

$\left(\mathrm{-8},\mathrm{-10}\right)$ and $\left(9,5\right)$

**Write the Equation of a Circle in Standard Form**

In the following exercises, write the standard form of the equation of the circle with the given information.

radius is 15 and center is $\left(0,0\right)$

radius is 9 and center is $\left(\mathrm{-3},5\right)$

center is $\left(3,6\right)$ and a point on the circle is $\left(3,\mathrm{-2}\right)$

**Graph a Circle**

In the following exercises, ⓐ find the center and radius, then ⓑ graph each circle.

$2{x}^{2}+2{y}^{2}=450$

${\left(x+3\right)}^{2}+{\left(y-5\right)}^{2}=81$

${x}^{2}+{y}^{2}-6x-12y-19=0$

##### Parabolas

**Graph Vertical Parabolas**

In the following exercises, graph each equation by using its properties.

$y={x}^{2}+4x-3$

$y=\mathrm{-6}{x}^{2}+12x-1$

In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.

$y={x}^{2}+4x+7$

$y=\mathrm{-3}{x}^{2}-18x-29$

**Graph Horizontal Parabolas**

In the following exercises, graph each equation by using its properties.

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

$x=\text{\u2212}{y}^{2}+2y-4$

In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.

$x=4{y}^{2}+8y$

$x=\text{\u2212}{y}^{2}-6y-7$

**Solve Applications with Parabolas**

In the following exercises, create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in standard form.

##### Ellipses

**Graph an Ellipse with Center at the Origin**

In the following exercises, graph each ellipse.

$\frac{{x}^{2}}{36}+\frac{{y}^{2}}{25}=1$

$49{x}^{2}+64{y}^{2}=3136$

**Find the Equation of an Ellipse with Center at the Origin**

In the following exercises, find the equation of the ellipse shown in the graph.

**Graph an Ellipse with Center Not at the Origin**

In the following exercises, graph each ellipse.

$\frac{{\left(x-1\right)}^{2}}{25}+\frac{{\left(y-6\right)}^{2}}{4}=1$

$\frac{{\left(x-5\right)}^{2}}{16}+\frac{{\left(y+3\right)}^{2}}{36}=1$

In the following exercises, ⓐ write the equation in standard form and ⓑ graph.

${x}^{2}+{y}^{2}+12x+40y+120=0$

$25{x}^{2}+4{y}^{2}+150x+125=0$

**Solve Applications with Ellipses**

In the following exercises, write the equation of the ellipse described.

A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 10 AU and the furthest is approximately 90 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.

##### Hyperbolas

**Graph a Hyperbola with Center at $\left(0,0\right)$**

In the following exercises, graph.

$\frac{{y}^{2}}{49}-\frac{{x}^{2}}{16}=1$

$16{x}^{2}-4{y}^{2}=64$

**Graph a Hyperbola with Center at $\left(h,k\right)$**

In the following exercises, graph.

$\frac{{\left(x-2\right)}^{2}}{4}-\frac{{\left(y-3\right)}^{2}}{16}=1$

$\frac{{\left(y-1\right)}^{2}}{25}-\frac{{\left(x-2\right)}^{2}}{9}=1$

In the following exercises, ⓐ write the equation in standard form and ⓑ graph.

$16{x}^{2}-4{y}^{2}-64x-24y-36=0$

$36{y}^{2}-16{x}^{2}-96x+216y-396=0$

**Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola**

In the following exercises, identify the type of graph.

ⓐ $16{y}^{2}-9{x}^{2}-36x-96y-36=0$

ⓑ ${x}^{2}+{y}^{2}-4x+10y-7=0$

ⓒ $y={x}^{2}-2x+3$

ⓓ $25{x}^{2}+9{y}^{2}=225$

ⓐ ${x}^{2}+{y}^{2}+4x-10y+25=0$

ⓑ ${y}^{2}-{x}^{2}-4y+2x-6=0$

ⓒ $x=-{y}^{2}-2y+3$

ⓓ $16{x}^{2}+9{y}^{2}=144$

##### Solve Systems of Nonlinear Equations

**Solve a System of Nonlinear Equations Using Graphing**

In the following exercises, solve the system of equations by using graphing.

$\{\begin{array}{c}y={x}^{2}-4\hfill \\ y=x-4\hfill \end{array}$

$\{\begin{array}{c}{x}^{2}+{y}^{2}=25\hfill \\ y=\mathrm{-5}\hfill \end{array}$

**Solve a System of Nonlinear Equations Using Substitution**

In the following exercises, solve the system of equations by using substitution.

$\{\begin{array}{c}{x}^{2}+{y}^{2}=4\hfill \\ x-y=4\hfill \end{array}$

$\{\begin{array}{c}{x}^{2}+4{y}^{2}=4\hfill \\ 2x-y=1\hfill \end{array}$

**Solve a System of Nonlinear Equations Using Elimination**

In the following exercises, solve the system of equations by using elimination.

$\{\begin{array}{c}{x}^{2}-{y}^{2}=5\hfill \\ -2{x}^{2}-3{y}^{2}=\mathrm{-30}\hfill \end{array}$

$\{\begin{array}{c}{x}^{2}+{y}^{2}=14\hfill \\ {x}^{2}-{y}^{2}=16\hfill \end{array}$

**Use a System of Nonlinear Equations to Solve Applications**

In the following exercises, solve the problem using a system of equations.

The sum of the squares of two numbers is 25. The difference of the numbers is 1. Find the numbers.

The difference of the squares of two numbers is 45. The difference of the square of the first number and twice the square of the second number is 9. Find the numbers.

The perimeter of a rectangle is 58 meters and its area is 210 square meters. Find the length and width of the rectangle.

Colton purchased a larger microwave for his kitchen. The diagonal of the front of the microwave measures 34 inches. The front also has an area of 480 square inches. What are the length and width of the microwave?