Question

The asymptotes of the hyperbola 6x- +13xy + 6y - 7x-8y - 26 = 0 are
(A)2x+3y-1=0, 3x +2y+2=0
(D)
(B) 2x+3y = 1, 3x + 2y = 2
(D) 2x+3y = 3, 3x +2y = 4
(0) 2x+3y = 0, 3x +2y = 0
mation of the hyperbola which passes through the​

Answer 1

$\textbf{Given:}$

$\text{Equation of hyperbola}$

$6x^2+13xy+6y^2-7x-8y-26=0$

$\textbf{To find:}$

$\text{Asymptotes of the given hyperbola}$

$\textbf{Solution:}$

$\text{We know that,}$

$\textbf{The combined equation of asymptotes and the equation}$

$\textbf{of hyperbola differ only by constant term}$

$\text{The combined equation of the asymptotes can be written as}$

$6x^2+13xy+6y^2-7x-8y+k=0$

$\text{Consider,}$

$6x^2+13xy+6y^2$

$=6x^2+9xy+4xy+6y^2$

$=3x(2x+3y)+2y(2x+3y)$

$=(3x+2y)(2x+3y)$

$\text{Now,}$

$6x^2+13xy+6y^2-7x-8y+k=(3x+2y+l)(2x+3y+m)$

$\text{Equating corresponding coefficients of x and y on bothsides}$

$2l+3m=-7$.........(1)

$3l+2m=-8$.........(2)

$(1){\times}2\implies\,4l+6m=-14$

$(2){\times}3\implies\,9l+6m=-24$

$\text{Subtracting, we get}$

$-5l=10$

$\implies\,l=-2$

$\text{Put l=-2 in (1)}$

$2(-2)+3m=-7$

$-4+3m=-7$

$3m=-7+4$

$3m=-3$

$\implies\,m=-1$

$\therefore\textbf{Equation of asymptotes are}$

$3x+2y-2=0$

$2x+3y-1=0$

$\textbf{Answer:}$

$\textbf{Option (B) is correct}$

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