Pair of Straight Lines

The equations of two or more lines can be expressed together by an equation of degree higher than one. As we see that a linear equation in x and y represents a straight line, the product of two linear equations represent two straight lines, that is a pair of straight lines. Hence we study pair of straight lines as a quadratic equations in x and y . Let L ≡ a x + b y + c = and L ≡ a x + b y + c = , be separate equations of two straight lines.

If P ( x , y ) is a point on L , then it satisfies the equaiton L = . Similarly, if P ( x , y ) is on L then L = . If P ( x , y ) lies either on L = or L = , then P ( x , y ) satisfies the equation ( L ) ( L ) = , and no other point satisfies L · L = . Therefore the equation L · L = represents the pair of straight lines L = and L = .

For example, consider the two equations y + x = and y − x = . The above two equations represent the equation of two straight lines passing through the origin with slopes − and respectively. Combining the above equation, we get y + x y − x = ⇒ y − x = , represents the pair of straight lines x = x = y − √ y + √ Pair of straight lines y − x = Figure . .

Pair of Straight Lines . . Pair of Lines Passing through the Origin We first consider a simple case. Both the lines in this pair pass through the origin.

Thus, their equations can be written as y − m x = y − m x = Combined equation of these two lines is ( y − m x ) ( y − m x ) = ( . ) y − ( m + m ) xy + m m x =