Show that the equations x²-2xy+y²-2x+2y=0 represents two straight lines which are parallel to each other. | Pair Of Straight Lines | SciPiPupil

Question: Show that the equations x²-2xy+y²-2x+2y=0 represents two straight lines which are parallel to each other.

Solution:

Given,

Single equation of two straight lines is $x² -2xy +y² -2x +2y = 0$

$or, x² -2xy +y² -2x +2y = 0$

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

$or, (x -y)² - 2(x -y)= 0$

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

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

So, the two separate equations of straight lines are $x -y = 0$ and $x -y -2 = 0$.

For line 1 (x -y = 0$),

Slope ($m_1$) =$-\frac{coefficient \; of \;x}{coefficient \; of \; y}$

$= - \frac{1}{-1}$

$= 1$

For line 2 (x -y -2 = 0),

Slope ($m_2$) = $- \frac{coefficient \; of x }{ coefficient \; of \; y}$

$= - \frac{1}{-1}$

$= 1$

For line lines to be parallel, their slopes should be equal.

$or, m_1 = m_2$

$or, 1 = 1$ which is true.

Therefore, it is proved that the two separate equations of lines represented by the given single equation are parallel to each other.

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