Ellipses & Hyperbolas.

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Presentation transcript:

Ellipses & Hyperbolas

Ellipse Definition The set of all points, the sum of whose distances from the foci is constant.

Examples Find the center, vertices, foci, and eccentricity of the ellipse, and sketch its graph. 𝑥 2 16 + 𝑦 2 81 =1

Examples Find the center, vertices, foci, and eccentricity of the ellipse, and sketch its graph. 𝑥+3 2 16 + 𝑦−2 2 12 =1

Examples Put the ellipse in standard form. Then find its center, vertices, foci, and eccentricity. Sketch a graph of the ellipse. 16 𝑥 2 + 𝑦 2 =16

Examples Put the ellipse in standard form. Then find its center, vertices, foci, and eccentricity. Sketch a graph of the ellipse. 9 𝑥 2 +4 𝑦 2 −54𝑥+40𝑦+37=0

Examples Put the ellipse in standard form. Then find its center, vertices, foci, and eccentricity. Sketch a graph of the ellipse. 𝑥 2 +4 𝑦 2 −6𝑥+20𝑦−2=0

Hyperbolas

Hyperbola Definition The set of all points for which the absolute value of the difference between the distances from the foci is constant.

Find the center, vertices, foci, asymptotes and graph the hyperbola Find the center, vertices, foci, asymptotes and graph the hyperbola. Find its eccentricity. 𝑥 2 9 − 𝑦 2 25 =1 𝑦 2 9 − 𝑥 2 =1

Find the center, vertices, foci, asymptotes, and graph the hyperbola. 𝑦−3 2 144 − 𝑥+1 2 25 =1 𝑥−1 2 81 − 𝑦 2 25 =1

Find the center, vertices, foci, asymptotes, and graph the hyperbola. 𝑥 2 −9 𝑦 2 +36𝑦−72=0 16 𝑦 2 − 𝑥 2 +2𝑥+64𝑦+63=0