Binomial Theorem

Question
CBSEENMA11015225

For the hyperbola 3X2 – 2y2 = 1.

Find: (i) the lengths of the transverse and conjugate axes.

(ii) the co-ordinates of foci.

(iii) the co-ordinates of vertices.

(iv) the eccentricity.

(v) the length of latus rectum.

Solution

Equation of the hyperbola is 3 straight x squared minus 2 straight y squared equals 1 space rightwards double arrow space space fraction numerator straight x squared over denominator open parentheses begin display style 1 third end style close parentheses end fraction minus fraction numerator straight y squared over denominator open parentheses begin display style 1 half end style close parentheses end fraction equals 1
which is of the standard form straight x squared over straight a squared minus straight y squared over straight b squared equals 1
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Also,                            straight c squared equals straight a squared plus straight b squared space rightwards double arrow space straight c squared equals 1 third plus 1 half equals 5 over 6 space rightwards double arrow space straight c space equals space square root of 5 over 6 end root
(i) Length of transverse axis equals 2 straight a space equals space 2 cross times fraction numerator 1 over denominator square root of 3 end fraction equals fraction numerator 2 over denominator square root of 3 end fraction equals fraction numerator 2 square root of 3 over denominator 3 end fraction
Length of conjugate axis = 2b = 2 cross times fraction numerator 1 over denominator square root of 2 end fraction equals square root of 2
(ii) The co-ordinates of foci are:
                      space space straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space open parentheses negative square root of 5 over 6 comma end root 0 close parentheses space space and space space straight S subscript 2 left parenthesis straight c comma space 0 right parenthesis space left right arrow space open parentheses square root of 5 over 6 end root comma space 0 close parentheses
(iii) The co-ordinates of vertices are:
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(iv) Let the eccentricity be e.
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rightwards double arrow                                                  straight e equals square root of 5 over 2 end root
(v) Length of latus rectum  = fraction numerator 2 straight b squared over denominator straight a end fraction equals 2 cross times 1 half cross times fraction numerator square root of 3 over denominator 1 end fraction equals square root of 3

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