or x – ax + a + y = x + ax + a or y = ax ( a > ). Fig . (a) to (d) Fig . MATHEMATICS Hence, any point on the parabola satisfies y = ax .
... ( ) Conversely, let P( x , y ) satisfy the equation ( ) PF x – a x – a ax = PB ... ( ) and so P( x , y ) lies on the parabola. Thus, from ( ) and ( ) we have proved that the equation to the parabola with vertex at the origin, focus at ( a , ) and directrix x = – a is y = ax .
Discussion In equation ( ), since a > , x can assume any positive value or zero but no negative value and the curve extends indefinitely far into the first and the fourth quadrants. The axis of the parabola is the positive x -axis. Similarly, we can derive the equations of the parabola s in: Fig . (b) as y = – ax , Fig .
(c) as x = ay , Fig . (d) as x = – ay , These four equations are known as standard equations of parabola s.