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Question

The asymptotes of a hyperbola having centre at the point (2,2) are parallel to the lines x+3y=0 and 2xy=0. If the hyperbola passes through the point (1,4), Find the equation of the hyperbola.

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Solution

Let the asymptotes be x+3y+c1=0 and 2xy+c2=0
Since, the asymptotes passes through the centre (2,2) of the hyperbola.
,26+c1=0 and 4+2+c2=0
c1=8,c2=2
Thus, the equations of the asymptotes are
x+3y+8=0 and 2xy+2=0
Let the equation of the hyperbola be
(x+3y+8)(2xy+2)+λ=0 ........(1)
It passes through (1,4).
,(112+8)(2+4+2)+λ=0
5×4+λ=0
20+λ=0
λ=20
Putting the value of λ in (1), we obtain
(x+3y+8)(2xy+2)+20=0
This is the equation of the required hyperbola.


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