A Note The standard equations of parabolas have focus on one of the coordinate axis; vertex at the origin and thereby the directrix is parallel to the other coordinate axis. However, the study of the equations of parabolas with focus at any point and any line as directrix is beyond the scope here. From the standard equations of the parabolas, Fig11. , we have the following observations: .
Parabola is symmetric with respect to the axis of the parabola.If the equation has a y term, then the axis of symmetry is along the x -axis and if the equation has an x term, then the axis of symmetry is along the y -axis. . When the axis of symmetry is along the x -axis the parabola opens to the (a) right if the coefficient of x is positive, (b) left if the coefficient of x is negative. .
When the axis of symmetry is along the y -axis the parabola opens (c) upwards if the coefficient of y is positive. (d) downwards if the coefficient of y is negative. CONIC SECTIONS . .
Latus rectum Definition Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola (Fig11. ). To find the Length of the latus rectum of the parabola y = ax (Fig . ).
By the definition of the parabola, AF = AC. But AC = FM = a Hence AF = a . And since the parabola is symmetric with respect to x -axis AF = FB and so AB = Length of the latus rectum = a . Fig .
Fig . Example Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of the parabola y = x . Solution The given equation involves y , so the axis of symmetry is along the x -axis. The coefficient of x is positive so the parabola opens to the right.
Comparing with the given equation y