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Q)

Find the equation of the hyperbola if the asymptotes are $2x+3y-8=0$ and $3x-2y+1=0$ and $ (5 , 3 ) $ is a point on the hyperbola.

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A)
Toolbox:
  • The equation of a hyperbola and the combined equation of the asymptotes differ only in the constant term.
Step 1:
The combined equation of the asymptotes is $(2x+3y-8)(3x-2y+1)=0$
The equation of the hyperbola differs only in the constant term.
Let the equation be $(2x+3y-8)(3x-2y+1)=k$
Step 2:
The hyperbola passes through $(5,3)$
Therefore $(10+9-8)(15-6+1)=k$
$\Rightarrow k=11\times 10=110$
The equation is $(2x+3y-8)(3x-2y+1)=110$
On simplifying we get,
$6x^2+5xy-6y^2-22x+19y+8=110$
$\Rightarrow 6x^2+5xy-6y^2-22x+19y-102=0$
This is the required equation .
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