find the equationof the hyperbola satifying the give conditions: Foci (4, 0), the latus rectum is of length 12.

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Latika Leekha · Answer

Let the equation of hyperbola is x 2 /a 2 &ndash; y 2 /b 2 = 1. where b 2 = a 2 (e 2 -1). For hyperbola, the eccentricity e > 1. The focus is given as (4, 0). Hence, we have ae = 4. This gives a = 4/e. &hellip; (1) Latus Rectum = 2b 2 / a = 12. Hence, b 2 = 6a. &hellip;... (2) since, b 2 = a 2 (e 2 -1) Using (1) and (2) in the above equation, we get 6a = a 2 (e 2 -1) 4/e = (4/e) 2 (e 2 -1) Simplifying, we get 4e 2 &ndash; 6e &ndash; 4 = 0 (2e +1)(e &ndash; 2) = 0. This gives e = -1/2 or 2. Hence, we have e = 2. ( since e > 1 in case of hyperbola). hence, we have a = 4/e = 4/2 = 2. Substituting this value we get b 2 = 6a = 6.2 = 12 Hence, the eqauation of hyperbola is x 2 /a 2 &ndash; y 2 /b 2 = 1 x 2 /4 &ndash; y 2 /12 = 1. Simplifying we get, the required hyperbola as 3x 2 &ndash; y 2 = 12. Thanks & Regards Latika Leekha askIITians Faculty

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find the equationof the hyperbola satifying the give conditions: Foci (4, 0), the latus rectum is of length 12.

find the equationof the hyperbola satifying the give conditions: Foci (4, 0), the latus rectum is of length 12.

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