Question

The equations of the asymptotes of a hyperbola are \( x+y+3=0 \), \( 2x-y+1=0 \). If \( (1,-2) \) is a point on this hyperbola, then the equation of its conjugate hyperbola is:

• A. \( 2x^2+xy-y^2+7x-2y-1=0 \)
• B. \( 2x^2+xy-y^2+7x-2y+13=0 \)
• C. \( 2x^2+xy+y^2-7x-2y-1=0 \)
• D. \( 2x^2+xy+y^2-7x-2y+13=0 \)

Hint:

For determining the equation of a hyperbola from its asymptotes, use the general form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \) and substitute known points to solve for coefficients.

Answer 1

The Correct Option is B

The equation of a hyperbola with given asymptotes can be written in the form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] Given that \( (1,-2) \) lies on the hyperbola, substituting and solving, we get: \[ 2x^2 + xy - y^2 + 7x - 2y + 13 = 0 \] Thus, the correct answer is: \[ 2x^2+xy-y^2+7x-2y+13=0 \]