Example 10 - 9x2 + 4y2 = 36, find foci, vertices, length - Ellipse - Defination

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Example 10 Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the eccentricity of the ellipse 9x2 + 4y2 = 36. Given 9x2 + 4y2 = 36 Dividing whole equation by 36 ﷐9﷐𝑥﷮2﷯ + 4﷐𝑦﷮2﷯﷮36﷯ = ﷐36﷮36﷯ ﷐9﷮36﷯x2 + ﷐4﷐𝑦﷮2﷯﷮36﷯ = 1 ﷐﷐𝑥﷮2﷯﷮4﷯ + ﷐﷐𝑦﷮2﷯﷮9﷯ = 1 Since 4 < 9 Hence the above equation is of the form ﷐﷐𝑥﷮2﷯﷮﷐𝑏﷮2﷯﷯ + ﷐﷐𝑦﷮2﷯﷮﷐𝑎﷮2﷯﷯ = 1 Comparing (1) & (2) We know that c = ﷐﷮a2−b2﷯ c = ﷐﷮9−4﷯ c = ﷐﷮𝟓﷯ Co-ordinate of foci = (0, ± c) = (0, ± ﷐﷮5﷯) So co-ordinates of foci (0, ﷐﷮5﷯), & (0, −﷐﷮5﷯) Vertices = (0, ± a) = (0, ± 3) So vertices are (0, 3) & (0, −3) Length of major axis = 2a = 2 × 3 = 6 Length of minor axis = 2b = 2 × 2 = 4 Eccentricity e = ﷐c﷮a﷯ = ﷐﷐﷮5﷯﷮3﷯ Length of latus rectum = ﷐2b2﷮a﷯ = ﷐2 × 4﷮3﷯ = ﷐8﷮3﷯

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