A parallelogram is formed by the lines $a x^{2} + 2 h x y + b y^{2} = 0$ and the lines through $(p, q)$ parallel to them and the equation of the diagonal of the parallelogram which doesn't pass through origin is $(λ x - p) (a p + h q) + (μ y - q) (h p + b q) = 0$ , then find the value of $λ^{3} + μ^{3}$ .

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Solution

The combined equation of $A B$ and $A D$ is
$S 1 = a x^{2} + 2 h x y + b y^{2} = 0$
Now equation of lines passing through $(p, q)$ and parallel to $S_{1}$ is
$S 2 = a (x - p)^{2} + 2 h (x - p) (y - q) + b (y - q)^{2} = 0$
Hence, the equation of diagonal $B D$ is $S_{1} - S_{2} = 0$
$a (- 2 x p + p^{2}) + 2 h (- p y - q x + p q) + b (- 2 q y + q^{2}) = 0$
$= - a p (2 x - p) - h q (2 x - p) - h p (2 y - q) - b q (2 y - q) = 0 (∵ 2 h p q$ is written as $h p q + h p q)$
Hence, diagonal of $B D$ is
$(2 x - p) (a p + h q) + (2 y - q) (h p + b q) = 0$
So, $λ = 2, μ = 2$
Therefore, $λ^{3} + μ^{3} = 2^{3} + 2^{3} = 8 + 8 = 16$

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