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

167

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

173

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

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

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

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

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

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

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

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

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

195

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

540

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

540

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

186

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

244

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

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

224

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

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

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

177

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

203

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

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

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

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

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

216

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

168

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)

275

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)

275

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)

199

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

211

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)

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)

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)

202

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)

202

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

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

152

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

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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

161

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

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

151

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

190

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

142

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

152

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

142

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

168

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

179

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

135

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

427

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.

214