Question

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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Binomial Theorem

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.
$straight x squared over 36 plus straight y squared over 16 equals 1$

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