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

250

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

239

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

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

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

197

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

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

179

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

228

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

564

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

228

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

235

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

738

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

738

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

227

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

303

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

291

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

291

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

231

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

231

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

205

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

250

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

243

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

243

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

311

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

311

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

306

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

282

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

210

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)

325

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)

325

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)

247

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

254

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)

312

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)

312

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)

254

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)

254

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

333

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

187

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

205

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

206

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

208

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

225

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

179

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

225

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

171

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

188

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

180

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

206

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

213

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

179

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

527

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.

260