= m ( h – x ). . If three points ( h, ), ( a, b ) and ( , k) lie on a line, show that k b . Consider the following population and year graph (Fig .
), find the slope of the line AB and using it, find what will be the population in the year ? . Various Forms of the Equation of a Line We know that every line in a plane contains infinitely many points on it. This relationship between line and points leads us to find the solution of the following problem: Fig .
STRAIGHT LINES How can we say that a given point lies on the given line? Its answer may be that for a given line we should have a definite condition on the points lying on the line. Suppose P ( x, y ) is an arbitrary point in the XY-plane and L is the given line. For the equation of L, we wish to construct a statement or condition for the point P that is true, when P is on L, otherwise false.
Of course the statement is merely an algebraic equation involving the variables x and y . Now, we will discuss the equation of a line under different conditions. . .
Horizontal and vertical lines If a horizontal line L is at a distance a from the x -axis then ordinate of every point lying on the line is either a or – a [Fig . (a)] . Therefore, equation of the line L is either y = a or y = – a . Choice of sign will depend upon the position of the line according as the line is above or below the y -axis.
Similarly, the equation of a vertical line at a distance b from the y -axis is either x = b or x = – b [Fig . (b)]. Fig . Example Find the equations of the lines parallel to axes and passing through (– , ).
Solution Position of the lines is shown in the Fig . . The y-