The asymptotes of a hyperbola have center at the point (1,2) and are parallel to the lines 2x+3y=0 and 3x+2y=0. If the hyperbola passes through the point (5,3), then its equation is
A
(3x+2y−8)(2x+3y+7)−338=0
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B
(2x+3y−8)(3x+2y−7)−154=0
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C
(3x+2y−8)(2x+3y−7)−156=0
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D
(2x+3y−7)(3x+2y+8)−348=0
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Solution
The correct option is B(2x+3y−8)(3x+2y−7)−154=0 The equation of asymptotes parallel to the lines 2x+3y=0 and 3x+2y=0 are 2x+3y+λ=0⋯(1) 3x+2y+μ=0⋯(2) It is given that these asmptotes pass through the center (1,2). Substituting the point (1,2) in (1) and (2), we get λ=−8.μ=−7
Thus, the combined equation of asymptotes is (2x+3y−8)(3x+2y−7)=0
Let the equation of the hyperbola be (2x+3y−8)(3x+2y−7)+ω=0 This hyperbola passes through (5,3) ⇒(2⋅5+3⋅3−8)(3⋅5+2⋅3−7)+ω=0 ⇒ω=−154
Hence, the equation of the hyperbola is (2x+3y−8)(3x+2y−7)−154=0