The asymptotes of a hyperbola are parallel to $2 x + 3 y = 0$ and $3 x + 2 y = 0$ . The equation of that hyperbola whose centre is at(1,2) and passing through (5, 3) is

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$\begin{aligned} (2 x + 3 y - 8) (3 x + 2 y - 7) - 154 = 0 \end{aligned}$

$\begin{aligned} (2 x - 3 y - 8) (3 x - 2 y - 7) - 154 = 0 \end{aligned}$

$(3 x - 2 y + 8) (2 x + 3 y - 7) - 154 = 0$

$(3 x + 2 y - 8) (2 x + 3 y - 7) - 154 = 0$

answer is A.

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Detailed Solution

Asymptotes are always passing through the centre of the hyperbola'

The asymptote parallel to $2 x + 3 y = 0 is 2 (x - 1) + 3 (y - 2) = 0 \Rightarrow 2 x + 3 y - 8 = 0$

The asymptote parallel to $3 x + 2 y = 0 is 3 (x - 1) + 2 (y - 2) = 0 \Rightarrow 3 x + 2 y - 7 = 0$

The equation of hyperbola is $(2 x + 3 y - 8) (3 x + 2 y - 7) = k$

Since hyperbola passes through $(5, 3) we get (10 + 9 - 8) (15 + 6 - 7) = k \Rightarrow k = 154$

The equation of the hyperbola is $(2 x + 3 y - 8) (3 x + 2 y - 7) - 154 = 0$

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