The equation of a hyperbola, conjugate to the hyperbola x 2 + 3 x y + 2 y 2 + 2 x + 3 y + 1 = 0 , is Options: A . &nbsp x2+3xy+2y2+2x+3y+1=0 B . &nbspx2+3xy+2y2+2x+3y+2=0 C . &nbsp x2+3xy+2y2+2x+3y+3=0 D . &nbspx2+3xy+2y2+2x+3y+4=0

The equation of a hyperbola, conjugate to the hyperbola x2+3xy+2y2+2x+3y+1=0, is

Question

The equation of a hyperbola, conjugate to the hyperbola

x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 1 = 0

, is

Options:

A . x2+3xy+2y2+2x+3y+1=0

B . x2+3xy+2y2+2x+3y+2=0

C . x2+3xy+2y2+2x+3y+3=0

D . x2+3xy+2y2+2x+3y+4=0

Answer: Option B
:
B
We know that,
Equation of hyperbola + Equation of conjugate
hyperbola = 2(Equation of asymptotes)

∴

Equation of conjugate hyperbola
= 2 Equation of asymptotes - Equation of hyperbola
C(x, y) = 2A(x, y) – H(x, y) …… (i)

∵

H (x, y) \equiv x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y = 0

∴

Equation of asymptotes is

x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + λ = 0

∴

Δ = 0, a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0, t h e n λ = 1

∴

A (x, y) \equiv x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 1 = 0

∴

C (x, y) \equiv x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 2 = 0

[from equation (i)]

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