Find the equation of the parabola with focus − ( and directrix x = . Parabola is open left and axis of symmetry as x -axis and vertex ( , ) . Then the equation of the required parabola is y − = − ⇒ y = - x . Example .
Find the equation of the parabola whose vertex is ( , and focus( , Given vertex A ( , and focus S ( , and the focal distance AS = . Parabola is open left and symmetric about the line parallel to x -axis. Then, the equation of the required parabola is y + = − ( ) ⇒ = − ⇒ = . Example .
Find the equation of the parabola with vertex ( - - , axis parallel to y -axis and passing through ( , ) . Since axis is parallel to y -axis the required equation of the parabola is x + = a y + ) . Since this passes through ( , ), we get ) = ⇒ a = . Then the equation of parabola is x ) = which on simplifying yields, − = .
Example . Find the vertex, focus, directrix, and length of the latus rectum of the parabola x −= . For the parabola, - = ⇒ = y + ⇒ x = y + + Fig. .
Fig. . x = S - A ( , ) S ( ,− ) A ( ,− ) O A (– ,– ) y' x' O - - Two Dimensional Analytical Geometry - II ⇒ x − = y + ) which is in standard form. Therefore, a = and the vertex is ( , , and the focus is .
Equation of directrix is −+ = y + + = y + = . Length of latus rectum