Obtain the equation of hyperbola whose equations of asymptotes are x + 2y + 3 = 0 and 3x + 4y + 5 = 0 and hyperbola passes through (1, -1).

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Solution

The correct option is A

Combined equation of asymptotes (x+2y+5)(3x+4y+5)=0

$3 x^{2} + 4 x y + 5 x + 6 x y + 8 y^{2} + 10 y + g x + 12 y + 15 = 0$

$3 x^{2} + 8 y^{2} + 10 x y + 14 x + 22 y + 15 = 0$

Equatiopn of hyperbola

only difference between equation of hyperbola and the equation of asymptotes is constant part.

equation of hyperbola is

$3 x^{2} + 8 y^{2} + 10 x y + 14 x + 22 y + λ = 0$ ......... (1)

Since it passes through the point(1,-1)

substituting the x=1 & y=-1 in equation (1)

$3 \times 1 + 8 \times 1 + 10 \times (1) (- 1) + 14 \times 1 + 22 \times (- 1) + λ = 0$

$3 + 8 - 10 + 14 - 22 + λ = 0$

$λ = 7$

equation of hyperbola is

$3 x^{2} + 8 y^{2} + 10 x y + 14 x + 22 y + 7 = 0$

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Similar questions

Find the equation of conjugate hyperbola whose equations of asymptotes are x + 2y + 3 = 0 and 3x + 4y + 5 = 0 and hyperbola passes through (1, -1).

x + 2 y + 3 = 0

3 x + 4 y + 5 = 0

(1, - 1)

(1, - 1)

x + 2 y + 3 = 0

3 x + 4 y + 5 = 0

The equation of the hyperbola whose asymptotes are the lines 3x – 4y + 7 = 0 and 4x + 3y + 1 = 0 and which passes through origin is

3 x - 4 y + 7 = 0

4 x + 3 y + 1 = 0

(3 x - 4 y + 7) (4 x + 3 y + 1) = 7

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