Binomial Theorem

Question
CBSEENMA11015224

Find the equation of a hyperbola whose foci are left parenthesis 0 comma plus-or-minus square root of 10 right parenthesis end root and which pass through point (2, 3).

Solution

The foci of the hyperbola are left parenthesis 0 comma plus-or-minus square root of 10 right parenthesis comma which lie on y-axis.
∴ The standard equation of the hyperbola is straight y squared over straight a squared minus straight x squared over straight b squared equals 1                                                  ...(i)
It passes through (2, 3) rightwards double arrow    9 over straight a squared minus 4 over straight b squared equals 1                                                                       ...(ii)
The foci are left parenthesis 0 comma plus-or-minus square root of 10 right parenthesis space left right arrow space left parenthesis 0 comma plus-or-minus straight c right parenthesis space rightwards double arrow space straight c space equals space square root of 10
But,                   straight c squared equals straight a squared plus straight b squared space rightwards double arrow space left parenthesis square root of 10 right parenthesis squared space equals space straight a squared plus straight b squared space rightwards double arrow space straight a squared plus straight b squared equals 10 space or space straight b squared equals 10 minus straight a squared          ...(iii)
Using (iii) in (ii), we have 9 over straight a squared minus fraction numerator 4 over denominator 10 minus straight a squared end fraction equals 1
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When space straight a squared equals 18 from (iii),  straight b squared equals 10 minus 18 space space rightwards double arrow space straight b squared equals negative 8 (not possible)
When straight a squared equals 5 comma space from space left parenthesis iii right parenthesis comma space straight b squared equals 10 minus 5 equals 5
∴                          straight a squared equals 5 comma space straight b squared equals 5
Hence, from (i), the equation of the hyperbola is: straight y squared over 5 minus straight x squared over 5 equals 1 or straight y squared minus straight x squared equals 5.

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