Example 14 - Chapter 11 Class 11 Conic Sections

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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 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