$E q u a t i o n o f p a i r o f l i n e s t h r o u g h (0, 0) a n d f o r \min g a n e q u i l a t e r a l t r i a n g l e w i t h t h e l i n e 2 x - 3 y + 4 = 0 i s$

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$23 x^{2} + 3 y^{2} - 8 x y = 0$

$23 x^{2} - 3 y^{2} + 48 x y = 0$

$23 x^{2} + 3 y^{2} - 16 x y = 0$

$23 x^{2} + 3 y^{2} + 48 x y = 0$

answer is D.

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Detailed Solution

$\begin{array}{l} G i v e n l i n e 2 x - 3 y + 4 = 0 \\ C o m p a r i n g w i t h l x + m y + n = 0 \\ l = 2, m = - 3, n = 4 \\ Re . E q u a t i o n i s {(l x + m y)}^{2} - 3 {(m x - l y)}^{2} = 0 \\ \Rightarrow {(2 x - 3 y)}^{2} - 3 {(- 3 x - 2 y)}^{2} = 0 \\ \Rightarrow 4 x^{2} + 9 y^{2} - 12 x y - 3 (9 x^{2} + 4 y^{2} + 12 x y) = 0 \\ \Rightarrow 4 x^{2} + 9 y^{2} - 12 x y - 27 x^{2} - 12 y^{2} - 36 x y = 0 \\ \Rightarrow - 23 x^{2} - 48 x y - 3 y^{2} = 0 \\ \Rightarrow 23 x^{2} + 48 x y + 3 y^{2} = 0 \end{array}$