A Note If the fixed point lies on the fixed line, then the set of points in the plane, which are equidistant from the fixed point and the fixed line is the straight line through the fixed point and perpendicular to the fixed line. We call this straight line as degenerate case of the parabola. A line through the focus and perpendicular to the directrix is called the axis of the parabola. The point of intersection of parabola with the axis is called the vertex of the parabola (Fig11.
). . . Standard equations of parabola The equation of a parabola is simplest if the vertex is at the origin and the axis of symmetry is along the x -axis or y -axis.
The four possible such orientations of parabola are shown below in Fig11. (a) to (d). CONIC SECTIONS We will derive the equation for the parabola shown above in Fig . (a) with focus at ( a , ) a > ; and directricx x = – a as below: Let F be the focus and l the directrix .
Let FM be perpendicular to the directrix and bisect FM at the point O. Produce MO to X. By the definition of parabola, the mid-point O is on the parabola and is called the vertex of the parabola. Take O as origin, OX the x -axis and OY perpendicular to it as the y -axis.
Let the distance from the directrix to the focus be a . Then, the coordinates of the focus are ( a , ), and the equation of the directrix is x + a = as in Fig11. . Let P( x , y ) be any point on the parabola such that PF = PB, ...
( ) where PB is perpendicular to l . The coordinates of B are (– a , y ). By the distance formula, we have PF = x – a and PB = Since PF = PB, we have x – a i.e. ( x – a ) + y = ( x + a )