# Basic Equations of Parabolas

As just discussed, a parabola is the locus of a moving point *P* such that its distance from a fixed point (the focus *F*) is always equal to its distance from a fixed line *L = 0*, the directrix.

For a start (and to make things easier), we assume *F* to be the point (*a, 0*) and the line *L* to be \(L \equiv x + a = 0.\) The origin then lies mid-way between *F* and *L*. There’s no loss of generality in doing so since howsoever *F* and *L* may lie in the plane, we can always (by a suitable choice of the axes) make the origin of our axes lie half-way between *F* and *L *and the y-axis parallel to *L*.

Observe that since the origin *O* is equidistant from *F* and *L = 0*, it too lies on the parabola. Thus, our parabola will pass through the origin.

To find the equation representing the parabola, we assume the co-ordinates of point *P* lying on it as (x, y). Thus,

By the definition of a parabola, these two distances must always be equal so that

\[\begin{align}& \qquad \;\; {(x - a)^2} + {y^2} = {(x + a)^2}\\&\Rightarrow\quad \fbox{\({y^2} = 4ax\)} \,\,\,:\,\,\,{\rm{Equation of a parabola}}\end{align}\]

This is referred to as the standard equation of the parabola with focus *F(a, 0)* and directrix \(L \equiv x + a = 0\) .

When we actually plot the curve \({y^2} = 4ax,\) we obtain the following shape. Any point on this curve is always equidistant from *F (a, 0)* and \(L \equiv x + a = 0\) .

The line *y = 0* which passes through the focus *F(a, 0)* as is perpendicular to the directrix *x + a = 0* will be termed the **axis** of the parabola. The origin, which is halfway between the focus and the directrix, will be termed the **vertex **of the parabola.

It should now be obvious for you to deduce what the curve \({y^2} = 4ax,\) will look like if *a < 0*. In this case, the focus *F(a, 0)* lies on the negative *x*-axis while the directrix *x + a = 0* is to the right of the origin:

We can also deduce what the curve \({x^2} = 4ay\) will look like; just interchange the role of *x* and *y*. The focus and directrix will change accordingly as described in the figure below

In fact, we can now generalise this discussion to a parabola with vertex at *V (h, k)* and the axis parallel to the *x*-axis or the *y*-axis. Note that in these parabolas, the focus lies at a distance of |a| from the vertex along the axis of the parabola :

**Example - 1**

Plot the parabola given by the equation \({y^2} - 4y + 4x - 4 = 0\) .

**Solution:** The given equation can be rearranged as

\[{(y - 2)^2} = - 4(x - 2)\]

This represents a parabola with vertex *V(2, 2)* and opening towards the left (the fourth case above in Fig - 10) because *a *= –1 (negative).

The focus will lie at a distance 1 unit to the left of (2, 2), i.e. ,at (1, 2). The directrix will lie 1 unit to the right of (2, 2), i.e. it will be x = 3.

The following figure shows this parabola: