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Standard Equations of a Parabola

To obtain the equation of a parabola in a simplified form, we take the origin as the vertex V , the x axis as axis of symmetry . Let us take VF = a, so that the coordinates of F are (a, 0). Since the directrix D is at a distance a from the vertex and is parallel to y-axis, its equation is x + a = 0.

Let P (x, y) be any point on the parabola. Let M be the foot of perpendicular from P to the directrix D. By definition PF = PM.

We have PF =
and PM = |x + a|


Since PF = PM, we get PF2 = PM2 (x - a)2 + y2 = (x + a) 2
y2 = (x + a) 2 - (x - a) 2 = 4ax y2 = 4ax.

Shape of Parabolic Curve: y2 = 4ax (a > 0)

  1. x cannot be negative, because for x < 0, y is imaginary. Thus, no part of parabola lies to the left of y-axis. That is, the entire curve lies to the right side of y-axis.
  2. The curve passes through the origin, since the origin (0, 0) satisfies its equation. Further, x = 0 y2 = 0,  so the y-axis touches the curve at (0, 0).
  3. From y2 = 4ax we get y = ± 2(ax)1/2. This means, corresponding to one positive value of x, there are two, equal and opposite values of y. This shows that the parabolic curve is symmetrical about the x-axis.
  4. As x increases, y also increases and there is no limit to this joint increase of x and y. That is, as x , y ± .
The sketch of y2 = 4ax is given in figure given below.

In the following table, we list the graph of four parabolas. Each of these parabolas has its vertex at the origin and the focus either on x-axis or y-axis. We shall take a > 0.








(a, 0)

x = -a

y2 = 4ax

Parabola, axis of symmetry is the x-axis, opens to the right



x = a

y2 = -4ax

Parabola, axis of symmetry is the x - axis, opens to the left

(0, 0)

(0, a)

y = -a

x2 = 4ay

Parabola, axis of symmetry is the y-axis, opens up

(0, 0)

(0, -a)

y = a

x2 = -4ay

Parabola, axis of symmetry is the y-axis, opens down

To derive the equation of tangents and normals, etc. we shall use the equation y2 = 4ax.

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