Binomial Theorem

Question
CBSEENMA11015162

In the following, find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
straight x squared over 36 plus straight y squared over 16 equals 1

Solution

Equation of ellipse is straight x squared over 36 plus straight y squared over 16 equals 1
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∴ Equation of ellipse is space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1
Now,                         space space space space straight a squared space equals space 36 space rightwards double arrow space space space straight a space equals space 6 space space and space space straight b squared equals 16 space rightwards double arrow space straight b space equals space 4
Co-ordinates of foci
Focus,          straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of straight a squared minus straight b squared end root comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of 36 minus 16 end root comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of 20 comma space 0 right parenthesis space left right arrow space left parenthesis negative 2 square root of 5 comma space 0 right parenthesis
Focus,            straight S subscript 2 space left parenthesis straight c comma space 0 right parenthesis left right arrow left parenthesis 2 square root of 5 comma space 0 right parenthesis
Co-ordinates of vertices
                       straight A space left parenthesis straight a comma space 0 right parenthesis space left right arrow space left parenthesis 6 comma space 0 right parenthesis
                     straight A apostrophe space left parenthesis negative straight a comma space 0 right parenthesis space left right arrow space left parenthesis negative 6 comma space 0 right parenthesis
Length of major axis = 2a = 2 (6) = 12
Length of minor axis = 2b = 2 x 4 = 8
Let the eccentricity be e. Now, c = ae rightwards double arrow 2 square root of 5 space equals space 6 straight e space rightwards double arrow space straight e space equals space fraction numerator square root of 5 over denominator 3 end fraction
∴                             straight e equals fraction numerator square root of 5 over denominator 3 end fraction
Length of latus rectum, l equals fraction numerator 2 straight b squared over denominator straight a end fraction space rightwards double arrow space l equals fraction numerator 2 cross times 16 over denominator 6 end fraction equals 16 over 3

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