Question

The equation of the hyperbola which passes through the origin and having the assymptotes as 3x - 4y + 7 = 0 and 4x + 3y + 1 = 0 is .

• A. $12x^2-7\mathrm{xy}-12y^2-31x+17y=0$
  
• B. $12x^2-7\mathrm{xy}-12y^2+31x-17y=0$
  
• C. $12x^2-7\mathrm{xy}-12y^2-31x-17y=0$
  
• D. $12x^2-7\mathrm{xy}-12y^2+31x+17y=0$

Answer 1

The correct option is D $12x^2-7\mathrm{xy}-12y^2+31x+17y=0$

The combined equation of assymptotes is
(3x - 4y + 7)(4x + 3y + 1 ) = 0
$\Rightarrow 12x^2-7\mathrm{xy}-12y^2+31x+17y+7=0$
Since the equation of a hyperbola and its assymptotes only differ by a constant term.
Hence, the equation of the hyperbola is:
$12x^2-7\mathrm{xy}-12y^2+31x+17y+k=0...\left(i\right)$
As (i) passes through Origin, we have k=0.
Hence, the equation of the hyperbola is:
$12x^2-7\mathrm{xy}-12y^2+31x+17y=0$