8. Conic Sections

Given the ellipse equation $\frac{x^2}{16}+\frac{y^2}{4}=1$ , determine the magnitude of the semi-major axis (a) and the semi-minor axis (b).

Determine the vertices and foci of the following ellipse: $\frac{x^2}{49}+\frac{y^2}{36}=1$ .

205

Determine the vertices and foci of the following ellipse: $\frac{x^2}{9}+\frac{y^2}{16}=1$ .

205

Find the standard form of the equation for an ellipse with the following conditions.

Foci = $\left(-5,0\right),\left(5,0\right)$

Vertices = $\left(-8,0\right),\left(8,0\right)$

166

Graph the ellipse $\frac{{(x - 1)}^{2}}{9} + \frac{{(y + 3)}^{2}}{4} = 1$ .

Determine the vertices and foci of the ellipse $\left(x+1\right)^2+\frac{\left(y-2\right)^2}{4}=1$ .

152

Graph the ellipse and locate the foci. (y^2)/25 + (x^2)/16 = 1

209

Find the standard form of the equation of the ellipse satisfying the given conditions. Foci: (-4,0), (4,0); Vertices: (-5,0) (5,0)

Find the standard form of the equation of the ellipse satisfying the given conditions. Major axis horizontal with length 12; length of minor axis = 4; center: (-3,5)

Graph the ellipse and locate the foci. 9x^2 + 4y^2 - 18x + 8y -23 = 0

505

Graph the ellipse and locate the foci. (y^2)/25 + (x^2)/16 = 1

209

Find the standard form of the equation of the ellipse satisfying the given conditions. Major axis horizontal with length 12; length of minor axis = 4; center: (-3,5)

In Exercises 1–18, graph each ellipse and locate the foci. x^2/16 +y^2/4 = 1

211

Graph the ellipse and locate the foci. (x^2)/36 +(y^2)/25 = 1

608

Graph the ellipse and locate the foci. (x^2)/36 +(y^2)/25 = 1

608

In Exercises 1–18, graph each ellipse and locate the foci. x^2/9 +y^2/36= 1

197

In Exercises 1–18, graph each ellipse and locate the foci. x^2/25 +y^2/64 = 1

270

In Exercises 1–18, graph each ellipse and locate the foci. x^2/49 +y^2/81 = 1

249

In Exercises 1–18, graph each ellipse and locate the foci. x^2/49 +y^2/81 = 1

249

In Exercises 1–18, graph each ellipse and locate the foci. x^2/(9/4) +y^2/(25/4) = 1

208

In Exercises 1–18, graph each ellipse and locate the foci. x^2/(9/4) +y^2/(25/4) = 1

208

In Exercises 1–18, graph each ellipse and locate the foci. x² = 1 – 4y²

187

In Exercises 1–18, graph each ellipse and locate the foci. 25x²+4y² = 100

217

In Exercises 1–18, graph each ellipse and locate the foci.4x²+16y² = 64

224

In Exercises 1–18, graph each ellipse and locate the foci.4x²+16y² = 64

224

In Exercises 1–18, graph each ellipse and locate the foci. 7x² = 35-5y²

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In Exercises 1–18, graph each ellipse and locate the foci. 7x² = 35-5y²

278

In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci.

258

In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci.

242

In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci.

181

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-5, 0), (5, 0); vertices: (-8, 0), (8,0)

296

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-5, 0), (5, 0); vertices: (-8, 0), (8,0)

296

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (0, -4), (0, 4); vertices: (0, −7), (0, 7)

218

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-2, 0), (2, 0); y-intercepts: -3 and 3

229

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length 8; length of minor axis = 4; center: (0, 0)

284

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length 8; length of minor axis = 4; center: (0, 0)

284

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Major axis vertical with length 10; length of minor axis = 4; center: (-2, 3)

223

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Major axis vertical with length 10; length of minor axis = 4; center: (-2, 3)

223

In Exercises 37–50, graph each ellipse and give the location of its foci. (x − 2)²/9 + (y -1)² /4= 1

277

In Exercises 37–50, graph each ellipse and give the location of its foci. (x +3)²+ 4(y -2)² = 16

167

In Exercises 37–50, graph each ellipse and give the location of its foci. (x − 4)²/9 + (y +2)² /25= 1

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In Exercises 37–50, graph each ellipse and give the location of its foci. x²/25 + (y -2)² /36= 1

180

In Exercises 37–50, graph each ellipse and give the location of its foci. (x +3)²/9 + (y -2)² = 1

176

In Exercises 37–50, graph each ellipse and give the location of its foci. (x − 1)²/2 + (y +3)² /5= 1

197

In Exercises 37–50, graph each ellipse and give the location of its foci. 9(x − 1)²+4(y+3)² = 36

160

In Exercises 49–56, identify each equation without completing the square. 4x^2 - 9y^2 - 8x - 36y - 68 = 0

201

In Exercises 51–60, convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 9x^2 +25y² - 36x + 50y – 164 = 0

153

In Exercises 51–60, convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 9x² + 16y² – 18x + 64y – 71 = 0

164

In Exercises 51–60, convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 4x² + y²+ 16x - 6y - 39 = 0

154

In Exercises 51–60, convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 25x²+4y² – 150x + 32y + 189 = 0

183

In Exercises 51–60, convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 36x^2 +9y^2 - 216x = 0

191

In Exercises 61–66, find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.

In Exercises 67–68, graph each semiellipse. y = √16 - 4x²

149

Find the standard form of the equation of an ellipse with vertices at (0, -6) and (0, 6), passing through (2, 4).

460

The equation of the red ellipse in the figure shown is x^2/25 + y^2/9 =1Write the equation for each circle shown in the figure.

234