For hyperbola whose center is at (1, 2) and the asymptotes are parallel to lines `2x+3y=0` and `x+2y=1` , the equation of the hyperbola passing through (2, 4) is (a) `(2x+3y-5)(x+2y-8)=40` (b) `(2x+3y-8)(x+2y-5)=40` (c) `(2x+3y-8)(x+2y-5)=30` (d) none of these

Options

A. `(2x+3y-5)(x+2y-8)=40`
B. `(2x+3y-8)(x+2y-5)=40`
C. `(2x+3y-8)(x+2y-5)=30`
D. none of these

Answer

B

Explanation

Let the asymptotes be `2x+3y+lambda_(1)=0 and x+2y+lambda_(2)=0`.
It will pass through the centre (1, 2). Hence,
`lambda_(2)=-8, lambda_(2)=-5`
The equation of the hyperbola is
`(2x+3y-8)(x+2y-5)+lambda=0`
It passes through (2, 4). Therefore,
`(4+12-8)(2+8-5)+lambda=0 or lambda=-40`
Hence, equation of hyperbola is
`(2x+3y-8)(x+2y-5)=40`