Show that the equation x²-5xy +4y² +2x = 8y represents two straight lines. | Pair Of Straight Lines | SciPiPupil

Question: Show that the equation x²-5xy +4y² +2x = 8y represents two straight lines.

Solution:

Given,

Single equation of two straight lines is $x² -5xy +4y² + 2x = 8y$

$or, x² -5xy +4y² = 8y -2x$

$or, x² - (4+1)xy +4y² = 2(4y -x)$

$or, x² -4xy -xy +4y² = 2(4y -x)$

$or, x(x -4y) - y (x -4y) = 2(4y -x)$

$or, (x -y)(x -4y) = 2(4y -x)$

$or, (x -y)(x -4y) - 2(4y -x) = 0$

$or, (x-y)(x -4y) + 2(x -4y) = 0$

$or, (x -4y)(x -y +2) = 0$

So, one of the line is $x -4y = 0$ and the other line is $x -y +2 = 0$.

Hence, it is proved that the given single equation of line represents two straight lines.

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