= ax , we find that a = . Thus, the focus of the parabola is ( , ) and the equation of the directrix of the parabola is x = – (Fig . ). Length of the latus rectum is a = × = .
Fig . MATHEMATICS Example Find the equation of the parabola with focus ( , ) and directrix x = – . Solution Since the focus ( , ) lies on the x -axis, the x -axis itself is the axis of the parabola. Hence the equation of the parabola is of the form either y = ax or y = – ax .
Since the directrix is x = – and the focus is ( , ), the parabola is to be of the form y = ax with a = . Hence the required equation is y = ( ) x = x Example Find the equation of the parabola with vertex at ( , ) and focus at ( , ). Solution Since the vertex is at ( , ) and the focus is at ( , ) which lies on y -axis, the y -axis is the axis of the parabola . Therefore, equation of the parabola is of the form x = ay .
thus, we have x = ( ) y , i.e., x = y. Example Find the equation of the parabola which is symmetric about the y -axis, and passes through the point ( ,– ). Solution Since the parabola is symmetric about y -axis and has its vertex at the origin, the equation is of the form x = ay or x = – ay , where the sign depends on whether the parabola opens upwards or downwards. But the parabola passes through ( ,– ) which lies in the fourth quadrant, it must open downwards.
Thus the equation is of the form x = – ay . Since the parabola passes through ( ,– ), we have = – a (– ), i.e., a