Find the equation to the pair of lines through the origin which are perpendicular to the lines represented by $2 x^{2} - 5 x y + y^{2} = 0$ .

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Solution

Two line represented by $2 x^{2} - 5 x y + 4 y^{2} = 0$ are as follows;
$x^{2} - 5 x y + 4 y^{2} = 0$
$x^{2} - 4 x y - x y + 4 y^{2} = 0$
$x (x - 4 y) - y (x - 4 y) = 0$
$(x - y) (x - 4 y) = 0$
$\to x = y$
$\to x = 4 y$

we know that to find perpendicular of any line we have to make,
slope of one is the opposite reciprocal of the slope of other.

For, $x = y$
slope of equation is $1$
so slope of perpendicular line will be $- 1$
and equation will be,
$\to x = (- y)$ which is perpendicular to the line $x = y$

For, $x = 4 y$
slope of equation is $4$
so slope of perpendicular line will be $- \frac{1}{4}$
and equation will be
$\to x = (- \frac{y}{4})$ which is perpendicular to the line $x = 4 y$

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