coordinate of every point on the line parallel to x- axis is , therefore, equation of the line parallel to x- axis and passing through (– , ) is y = . Similarly, equation of the line parallel to y -axis and passing through (– , ) is x = – . Fig . MATHEMATICS .
. Point-slope form Suppose that P ( x , y ) is a fixed point on a non-vertical line L, whose slope is m . Let P ( x, y ) be an arbitrary point on L (Fig . ).
Then, by the definition, the slope of L is given by m m , i.e. , ...( ) Since the point P ( x , y ) along with all points ( x, y ) on L satisfies ( ) and no other point in the plane satisfies ( ). Equation ( ) is indeed the equation for the given line L. Thus, the point ( x, y ) lies on the line with slope m through the fixed point ( x , y ), if and only if, its coordinates satisfy the equation y – y = m ( x – x ) Example Find the equation of the line through (– , ) with slope – .
Solution Here m = – and given point ( x , y ) is (– , ). By slope-intercept form formula ( ) above, equation of the given line is y – = – ( x + ) or x + y + = , which is the required equation. . .
Two-point form Let the line L passes through two given points P ( x , y ) and P ( x , y ). Let P ( x, y ) be a general point on L (Fig . ). The three points P , P and P are collinear, therefore, we have slope of P P = slope of P P i.e.,