In a Hyperbola - The Triangle formed by an latus rectam with its farthest vertex is equilateral then find its ecentricity

Asked by Malavika Umesh | 1st May, 2015, 01:06: PM

Expert Answer:

Consider equation of a hyperbola in standad form.
x squared over a squared minus y squared over b squared equals 1
Eccentricity 'e' is given by 
e squared equals 1 plus b squared over a squared..... left parenthesis 1 right parenthesis
The coodinates of ends of latus rectum of one half of the hyperbola are open parentheses a e comma space plus-or-minus b squared over a close parentheses
The coordinates of the vertex farther from the latus rectum are open parentheses negative a comma space 0 close parentheses
Since the triangle formed by these points is an equilateral triangle, equating the side lengths formed by these vertices we get
open parentheses a e plus a close parentheses squared plus open parentheses b squared over a close parentheses squared equals open parentheses fraction numerator 2 b squared over denominator a end fraction close parentheses squared rightwards double arrow open parentheses a e plus a close parentheses squared equals 3 open parentheses b squared over a close parentheses squared rightwards double arrow a e plus a equals square root of 3 open parentheses b squared over a close parentheses rightwards double arrow e plus 1 equals square root of 3 open parentheses b squared over a squared close parentheses
Adding square root of 3 to both sides of the above equation, we get 
e plus 1 plus square root of 3 equals square root of 3 open parentheses 1 plus b squared over a squared close parentheses.... left parenthesis 2 right parenthesis
From (1) and (2), we get 
square root of 3 e squared equals e plus 1 plus square root of 3 rightwards double arrow square root of 3 e squared minus e minus open parentheses 1 plus square root of 3 close parentheses equals 0 rightwards double arrow e equals fraction numerator 1 plus-or-minus square root of 1 plus 4 square root of 3 open parentheses 1 plus square root of 3 close parentheses end root over denominator 2 square root of 3 end fraction
Taking only the posiive sign, as 'e' can not be negative, we get
e equals fraction numerator 1 plus open parentheses 1 plus 2 square root of 3 close parentheses over denominator 2 square root of 3 end fraction equals fraction numerator 1 plus square root of 3 over denominator square root of 3 end fraction


Answered by satyajit samal | 3rd May, 2015, 10:04: AM

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