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Question

The asymptotes of a hyperbola having centre at the point (1,2) are parallel to the lines 2x+3y=0 and 3x+2y=0. If the hyperbola passes through the point (5,3), show that is equation is (2x+3y8)(3x+2y+7)=154.

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Solution

Let the asymptotes be 2x+3y+c1=0 and 3x+2y+c2=0
Since, the asymptotes passes through the centre (1,2) of the hyperbola.
,2+6+c1=0 and 3+4+c2=0
c1=8,c2=7
Thus, the equations of the asymptotes are
2x+3y8=0 and 3x+2y7=0
Let the equation of the hyperbola be
(2x+3y8)(3x+2y7)+λ=0 ........(1)
It passes through (5,3).
,(10+98)(15+67)+λ=0
,11×14+λ=0
λ=154
Putting the value of λ in (1), we obtain
(2x+3y8)(3x+2y7)154=0
This is the equation of the required hyperbola.

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