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Example 14 - Find foci, vertices, eccentricity, latus rectum - Hyperbola

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  1. Chapter 11 Class 11 Conic Sections
  2. Serial order wise
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Example 14 Find the coordinates of the foci and the vertices, the eccentricity, the length of the latus rectum of the hyperbolas: (i) ﷐x2﷮9﷯ - ﷐y2﷮16﷯ = 1, The given equation is ﷐𝑥2﷮9﷯ − ﷐𝑦2﷮16﷯ = 1 The above equation is of the form ﷐𝑥2﷮𝑎2﷯ − ﷐𝑦2﷮𝑏2﷯ = 1 Comparing (1) & (2) a2 = 9 a = 3 & b2 = 16 b = 4 Also, c2 = a2 + b2 c2 = 9 + 16 c2 = 25 c = 5 So, Co-ordinate of foci = (±c, 0) = (±5, 0) Thus, Co-ordinate of foci are (5, 0) & (−5, 0) Vertices = (±a, 0) = (±3, 0) Thus, Vertices are (3, 0) & (−3, 0) Eccentricity e = ﷐𝑐﷮𝑎﷯ = ﷐5﷮3﷯ The latus rectum = ﷐2𝑏2﷮𝑎﷯ = ﷐2 × 16﷮3﷯ = ﷐32﷮3﷯ Example 14 Find the coordinates of the foci and the vertices, the eccentricity, the length of the latus rectum of the hyperbolas: (ii) y2 – 16x2 = 16 The given equation is y2 – 16x2 = 16 Divide whole equation by 16 ﷐𝑦2−16𝑥2﷮16﷯ = ﷐16﷮16﷯ ﷐y2﷮16﷯ − ﷐x2﷮1﷯ = 1 The above equation is of the form ﷐﷐𝑦﷮2﷯﷮﷐𝑎﷮2﷯﷯ − ﷐﷐𝑥﷮2﷯﷮﷐𝑏﷮2﷯﷯ = 1 Comparing (1) & (2) a2 = 16 a = 4 & b2 = 1 b = 1 Also , c2 = a2 + b2 c2 = 16 + 1 c2 = 17 c = ﷐﷮𝟏𝟕﷯ Co-ordinate of foci = (0, ±c) = (0, ±﷐﷮17﷯) So, co-ordinates of foci are (0, ﷐﷮17﷯) & (0, −﷐﷮17﷯) Vertices = (0, ±a) = (0, ±4) So, vertices are (0, 4) & (0, −4) Eccentricity e = ﷐𝑐﷮𝑎﷯ = ﷐﷐﷮17﷯﷮4﷯ Latus rectum = ﷐2𝑏2﷮𝑎﷯ = ﷐2 × 1﷮4﷯ = ﷐1﷮2﷯

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