Eliminate $m, n$ from the equations $m^{2} x - m y + a = 0, n^{2} x - n y + a = 0, m n + 1 = 0$ .

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Solution

$m^{2} x - m y + a = 0$ ....... $(i)$
$n^{2} x - n y + a = 0$ ....... $(i i)$
Multiply both equations, we get
$(m^{2} x - m y + a) (n^{2} x - n y + a) = 0$
$(m^{2} n^{2} x^{2} - m^{2} n x y + a m^{2} x - m n^{2} x y + m n y^{2} - m a y + a n^{2} x - a n y + a^{2}) = 0$
As $m n + 1 = 0$ , we get
$(x^{2} + m x y + n x y + a m^{2} x - y^{2} - m a y + a n^{2} x - a n y + a^{2}) = 0$
Since, $x (m^{2} - n^{2}) - y (m - n) = 0$ ....... [Subtracting $(i i)$ from $(i)$ ]
$x (m + n) (m - n) - y (m - n) = 0$
$m + n = \frac{y}{x}$
Substitute $y = (m + n) x$ in the equation, we get
$x^{2} + m x (m + n) x + n x (m + n) x + a m^{2} x - (m + n)^{2} x^{2} - m a (m + n) x + a n^{2} x - a n (m + n) x + a^{2} = 0$
On simplification, we get
$x + a = 0$

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