. . Parabola Since e = , for a parabola, we note that the parabola is the locus of points in a plane that are equidistant from both the directrix and the focus. (i) Equation of a parabola in standard form with vertex at ( , ) Let S be the focus and l be the directrix.
Draw SZ perpendicular to the line l . Let us assume SZ produced as x -axis and the perpendicular bisector of SZ produced as y - axis. The intersection of this perpendicular bisector with SZ be the origin O . Fig.
. Fig. . P x y ( , ) M S x ( , l Directrix Focus l O ( , ) Z M Axis L Vertex L Latus Rectum S a ( , ) P( x , y ) - - Let SZ = .
Then S is( , ) a and the equation of the directrix is x = . Let P x y ( , ) be the moving point in the locus that yield a parabola. Draw PM perpendicular to the directrix. By definition, e = SP PM = .
So, SP PM Then, ( . On simplifying, we get y ax which is the equation of the parabola in the standard form . The other standard forms of parabola are y ax x ay , and x ay Definition . ● The line perpendicular to the directrix and passing through the focus is known as the Axis of the parabola.
● The intersection point of the axis with the curve is called vertex of the parabola ● Any chord of the parabola, through its focus is called focal chord of the parabola ● The length of the focal chord perpendicular to the axis is called latus rectum of the parabola Example . Find the length of Latus rectum of the parabola y ax Equation of the parabola is y ax Latus rectum LL ′ passes through the focus ( , ) a . Refer (Fig. .
) Hence the point L is ( ,