11.8.1: Review Exercises

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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.

244.

$(−5, 1)$ and $(−1, 4)$

245.

$(−2, 5)$ and $(1, 5)$

246.

$(8, 2)$ and $(−7, −3)$

247.

$(1, −4)$ and $(5, −5)$

Use the Midpoint Formula

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

248.

$(−2, −6)$ and $(−4, −2)$

249.

$(3, 7)$ and $(5, 1)$

250.

$(−8, −10)$ and $(9, 5)$

251.

$(−3, 2)$ and $(6, −9)$

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.

252.

radius is 15 and center is $(0, 0)$

253.

radius is $\sqrt{7}$ and center is $(0, 0)$

254.

radius is 9 and center is $(−3, 5)$

255.

radius is 7 and center is $(−2, −5)$

256.

center is $(3, 6)$ and a point on the circle is $(3, −2)$

257.

center is $(2, 2)$ and a point on the circle is $(4, 4)$

Graph a Circle

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

258.

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

259.

$3 x^{2} + 3 y^{2} = 432$

260.

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

261.

${(x + 2)}^{2} + {(y + 5)}^{2} = 49$

262.

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

263.

$x^{2} + y^{2} - 4 y - 60 = 0$

Parabolas

Graph Vertical Parabolas

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

264.

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

265.

$y = 2 x^{2} + 10 x + 7$

266.

$y = −6 x^{2} + 12 x - 1$

267.

$y = − x^{2} + 10 x$

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

268.

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

269.

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

270.

$y = −3 x^{2} - 18 x - 29$

271.

$y = − x^{2} + 12 x - 35$

Graph Horizontal Parabolas

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

272.

$x = 2 y^{2}$

273.

$x = 2 y^{2} + 4 y + 6$

274.

$x = − y^{2} + 2 y - 4$

275.

$x = −3 y^{2}$

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

276.

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

277.

$x = y^{2} + 4 y + 5$

278.

$x = − y^{2} - 6 y - 7$

279.

$x = −2 y^{2} + 4 y$

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.

280.

$The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 5 feet high and 20 feet wide.$

281.

$The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 25 feet high and 30 feet wide.$

Ellipses

Graph an Ellipse with Center at the Origin

In the following exercises, graph each ellipse.

282.

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

283.

$\frac{x^{2}}{4} + \frac{y^{2}}{81} = 1$

284.

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

285.

$9 x^{2} + y^{2} = 9$

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.

286.

$The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 10, 0), and co-vertices at (0, plus or minus 4).$

287.

$The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 8), and co-vertices at (plus or minus 6, 0).$

Graph an Ellipse with Center Not at the Origin

In the following exercises, graph each ellipse.

288.

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

289.

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

290.

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

291.

$\frac{{(x + 3)}^{2}}{9} + \frac{{(y - 2)}^{2}}{25} = 1$

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

292.

$4 x^{2} + 16 y^{2} + 48 x + 160 y + 480 = 0$

293.

$25 x^{2} + 4 y^{2} - 150 x - 56 y + 321 = 0$

294.

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

295.

$4 x^{2} + 9 y^{2} - 126 y + 405 = 0$

Solve Applications with Ellipses

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

296.

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.

$The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 50, 0), the sun marked as a foci and labeled (50, 0), the closest distance the comet is from the sun marked as 10 A U, and the farthest a comet is from the sun marked as 90 A U.$

Hyperbolas

Graph a Hyperbola with Center at $(0, 0)$

In the following exercises, graph.

297.

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

298.

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

299.

$9 y^{2} - 16 x^{2} = 144$

300.

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

Graph a Hyperbola with Center at $(h, k)$

In the following exercises, graph.

301.

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

302.

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

303.

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

304.

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

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

305.

$4 x^{2} - 16 y^{2} + 8 x + 96 y - 204 = 0$

306.

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

307.

$4 y^{2} - 16 x^{2} + 32 x - 8 y - 76 = 0$

308.

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

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

In the following exercises, identify the type of graph.

309.

ⓐ $16 y^{2} - 9 x^{2} - 36 x - 96 y - 36 = 0$
ⓑ $x^{2} + y^{2} - 4 x + 10 y - 7 = 0$
ⓒ $y = x^{2} - 2 x + 3$
ⓓ $25 x^{2} + 9 y^{2} = 225$

310.

ⓐ $x^{2} + y^{2} + 4 x - 10 y + 25 = 0$
ⓑ $y^{2} - x^{2} - 4 y + 2 x - 6 = 0$
ⓒ $x = - y^{2} - 2 y + 3$
ⓓ $16 x^{2} + 9 y^{2} = 144$

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.

311.

${\begin{matrix} 3 x^{2} - y = 0 \\ y = 2 x - 1 \end{matrix}$

312.

${\begin{matrix} y = x^{2} - 4 \\ y = x - 4 \end{matrix}$

313.

${\begin{matrix} x^{2} + y^{2} = 169 \\ x = 12 \end{matrix}$

314.

${\begin{matrix} x^{2} + y^{2} = 25 \\ y = −5 \end{matrix}$

Solve a System of Nonlinear Equations Using Substitution

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

315.

${\begin{matrix} y = x^{2} + 3 \\ y = −2 x + 2 \end{matrix}$

316.

${\begin{matrix} x^{2} + y^{2} = 4 \\ x - y = 4 \end{matrix}$

317.

${\begin{matrix} 9 x^{2} + 4 y^{2} = 36 \\ y - x = 5 \end{matrix}$

318.

${\begin{matrix} x^{2} + 4 y^{2} = 4 \\ 2 x - y = 1 \end{matrix}$

Solve a System of Nonlinear Equations Using Elimination

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

319.

${\begin{matrix} x^{2} + y^{2} = 16 \\ x^{2} - 2 y - 1 = 0 \end{matrix}$

320.

${\begin{matrix} x^{2} - y^{2} = 5 \\ - 2 x^{2} - 3 y^{2} = −30 \end{matrix}$

321.

${\begin{matrix} 4 x^{2} + 9 y^{2} = 36 \\ 3 y^{2} - 4 x = 12 \end{matrix}$

322.

${\begin{matrix} x^{2} + y^{2} = 14 \\ x^{2} - y^{2} = 16 \end{matrix}$

Use a System of Nonlinear Equations to Solve Applications

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

323.

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

324.

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.

325.

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

326.

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?

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Distance and Midpoint Formulas; Circles