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

211

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

210

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

517

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

215

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

620

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

620

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

199

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

276

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

257

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

257

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

213

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

213

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

220

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

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

226

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

280

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

265

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

250

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

186

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)

301

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)

301

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)

222

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

233

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)

289

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)

289

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

286

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

169

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

176

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

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

180

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

201

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

163

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

205

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

158

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

186

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

194

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).

471

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.

238