Question

Find the co-ordinates of foci, the vertices, the lenghts of transverse and conjugate axis, the eccentricity the length of latus rectum of hyperbola
$49 straight y squared minus 16 straight x squared equals 784$

Solution

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which is the equation of hyperbola in the standard form $straight y squared over straight a squared minus straight x squared over straight b squared equals 1$
∴ The foci and vertices lie on y-axis.
So, $space space straight a squared equals 16 space rightwards double arrow space straight a space equals space 4 space and space straight b squared equals 49 space rightwards double arrow space straight b equals 7$
Also, $straight c squared equals straight a squared plus straight b squared equals 16 plus 49 equals 65 space rightwards double arrow space straight c equals square root of 65$
The co-ordinates of foci are:
 $straight S subscript 1 left parenthesis 0 comma space minus straight c right parenthesis space left right arrow space left parenthesis 0 comma space minus square root of 65 right parenthesis space space space straight S subscript 2 left parenthesis 0 comma space straight c right parenthesis space left right arrow space left parenthesis 0 comma space square root of 65 right parenthesis$
The co-ordinates of vetices are:
 $straight A space left parenthesis 0 comma space straight a right parenthesis space left right arrow space left parenthesis 0 comma space 4 right parenthesis$
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The length of transverse axis = 2a = 2 x 4 = 8
The length of conjugate axis = 2b = 2 x 7 = 14
Let the eccentricity be e
∴ c = ae $rightwards double arrow space square root of 65 space equals space 4 straight e space rightwards double arrow space straight e space equals space fraction numerator square root of 65 over denominator 4 end fraction$
Length of latus rectum = $fraction numerator 2 straight b squared over denominator straight a end fraction space equals space fraction numerator 2 cross times 49 over denominator 4 end fraction equals 49 over 2$

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Binomial Theorem

Find the co-ordinates of foci, the vertices, the lenghts of transverse and conjugate axis, the eccentricity the length of latus rectum of hyperbola
$49 straight y squared minus 16 straight x squared equals 784$

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