Eliminate $x$ from the equations $a x^{2} - 2 a^{2} x + 1 = 0, a^{2} + x^{2} - 3 a x = 0$ .

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Solution

$a^{2} + x^{2} - 2 a x - a x = 0$
$(x - a)^{2} = a x$ ..... $(i)$
$a x^{2} - 2 a^{2} x + 1 = 0$ ..... $(i i)$
Divide this equation by a on both sides, we get
$x^{2} - 2 a x + \frac{1}{a} = 0$
Add both sides $a^{2},$ we get
$x^{2} - 2 a x + a^{2} + \frac{1}{a} = a^{2}$
$(x - a)^{2} = a^{2} - \frac{1}{a}$
$a x = a^{2} - \frac{1}{a}$ ........ From
$x = \frac{a^{3} - 1}{a^{2}}$
Substitute $a$ in the equation $a^{2} + x^{2} - 3 a x = 0$
We get on simplification
$a^{6} - a^{3} = 1$

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