Find x / y when x^2+6 y^2=5 x y - Vidyadip

Question

Find $x / y$ when $x^2+6 y^2=5 x y$

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Answer

$x^2+6 y^2=5 x y$
Dividing both sides by $y^2$ we get
$\frac{x^2}{y^2}+\frac{6 y^2}{y^2}=\frac{5 x y}{y^2}$
$\left(\frac{x}{y}\right)^2+6=5\left(\frac{x}{y}\right) $
$ \left(\frac{x}{y}\right)^2-5\left(\frac{x}{y}\right)+6=0$
$\text { Let } \frac{x}{y}=a $
$a^2-5 a+6=0$
$\Rightarrow (a-2)(a-3)=0 $
$ \Rightarrow a=2,3 $
$ \text { hence } \frac{x}{y}=2,3$

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