General equation of a parabola:

Let (h, k) be the focus S and lx + my + n = 0 the equation of the directrix ZM of a parabola. Let (x,y) be the coordinates of any point P on parabola. Then the relation, PS = distance of P from ZM, gives

(x - h)² + (y - k)² = (lx + my + n)² / (l² + m²)

This is general equation of the parabola.

Note:  From the general equation of parabola it is clear that 2^nd-degree terms in equation of a parabola form a perfect square. The converse is also true, that is if in an equation of the 2^nd degree, the second-degree terms form a perfect square then the equation represents a parabola, unless it represents two parallel straight lines. 

Special case:

Let vertex be (a, b) and axis be parallel to x-axis. Then the equation of parabola can be given by (y - b)² = 4a (x - a) which is equivalent to x = Ay² + By + C

Likewise, when the axis is parallel to y-axis and vertex be(a, b), the equation of parabola can be given by (x - α)² = 4a (y - β) which is equivalent to  y = A'x² + B'x + C'

Illustration: Find vertex, axis, directrix, tangent at vertex and the length of latus rectum of the parabola 2y² + 3y - 4x - 3 = 0.

Solution:             The given equation can be re written as 

                              which is of form Y² = 4aX.

                              Hence vertex is 

                              The axis is y+3/4=0      => y = -3/4

                              The directrix is X + a = 0

                              => x + 33/32 + 1/2 = 0 => x = -49/32

                              The tangent at vertex is x + 33/32 => x = -33/32

                              Length of latus rectum = 4a = 2.          

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