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: $\frac{x^2}{49}+\frac{y^2}{36}=1$ .

208

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

209

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

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

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

168

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

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

154

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

210

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

514

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

210

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

214

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

616

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

616

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

198

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

275

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

253

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

253

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

211

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

211

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

189

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

219

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

225

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

225

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

280

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

280

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

264

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

247

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

184

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)

300

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)

300

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)

221

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

232

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)

288

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)

288

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)

228

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)

228

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

283

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

168

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

175

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

183

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

179

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

199

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

162

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

203

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

155

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

166

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

157

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

185

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

193

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

151

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

469

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.

236