# Different Types of Parabola Equations

Here, you will learn Different Types of Parabola and Standard equations of parabola, focal chord, double ordinate and latus rectum of parabola.

Let’s begin –

## What is Parabola ?

A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point(focus) is equal to its perpendicular distance from a fixed straight line(directrix).

The Standard equation of parabola is $$y^2 = 4ax$$ and it is shown in figure. For this parabola :

(i) Vertex is (0,0).

(ii) focus is (a,0)

(iii) Axis is y = 0

(iv) Directrix is x + a = 0

(a) Focal distance :

The distance of a point on the parabola from the focus is called the focal distance of the point.

(b) Focal chord :

A chord of the parabola, which passes through the focus is called a focal chord.

(c) Double ordinate :

A chord of the parabola perpendicular to the axis of the symmetry is called double ordinate.

(d) Latus rectum :

A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is called latus rectum.

For $$y^2 = 4ax$$.

Length of the latus rectum = 4a

Length of the semi latus rectum = 2a

Ends of the latus rectum are L(a, 2a) & L'(a, -2a).

Note :

(i) Perpendicular distance from focus on the directrix = half the latus rectum.

(ii) Vertex is middle point of the focus & point of intersection of directrix & axis.

(iii) Two parabolas are said to be equal if they have the same latus rectum.

## Different Types of Parabola & Standard Equations of Parabola

Four different types of parabola equations are

$$y^2$$ = 4ax ; $$y^2$$ = -4ax ; $$x^2$$ = 4ay ; $$x^2$$ = -4ay.

One I had shown above and three others are shown below.

$$y^2$$ = -4ax

$$x^2$$ = 4ay

$$x^2$$ = -4ay

Parabola Vertex Focus Axis Directrix
$$y^2$$ = 4ax (0,0) (a,0) y = 0 x = -a
$$y^2$$ = -4ax (0,0) (-a,0) y = 0 x = a
$$x^2$$ = +4ay (0,0) (0,a) x = 0 y = -a
$$x^2$$ = -4ay (0,0) (0,-a) x = 0 y = a
$$(y-k)^2$$ = 4a(x-h) (h,k) (h+a,k) y = k x+a-h = 0
$$(x-p)^2$$ = 4b(y-q) (p,q) (p,b+q) x = p y+b-q = 0

Length of Latus rectum Ends of Latus rectum Parametric equation Focal length
4a (a,$$\pm$$2a) (a$$t^2$$, 2at) x + a
4a (-a,$$\pm$$2a) (-a$$t^2$$, 2at) x – a
4a ($$\pm$$2a,a) (2at, a$$t^2$$) y + a
4a ($$\pm$$2a,-a) (2at, -a$$t^2$$) y – a
4a (h+a, k$$\pm$$2a) (h+a$$t^2$$, k+2at) x – h + a
4b (p$$\pm$$2a, q+a) (p+2at, q+a$$t^2$$) y – q + b

Example : Find the vertex, axis, directrix, focus, latus rectum and the tangent at vertex for the parabola $$9y^2 – 16x – 12y – 57$$ = 0.

Solution : The given equation can be written as $$({y-2\over 3})^2$$ = $$16\over 9$$$$({x + 61\over 16})$$ which is of the form $$y^2$$ = 4ax. Hence the vertex is (-$$61\over 16$$, $$2\over 3$$)

The axis is y – $$2\over 3$$ = 0 $$\implies$$ y = $$2\over 3$$

The directrix is x + a – h = 0 $$\implies$$ x + $$61\over 16$$ + $$4\over 9$$ $$\implies$$ x = $$-613\over 144$$

The focus is (h+a, k) $$\implies$$ ($$-485\over 144$$, $$2\over 3$$)

Length of the latus rectum = 4a = $$16\over 9$$

The tangent at the vertex is x – h = 0 $$\implies$$ x = $$-61\over 16$$

Position of a point relative to a parabola :

The point ($$x_1$$,$$y_1$$) lies outside, on or inside the parabola $$y^2$$ = 4a$$x_1$$ is positive, zero or negative.

### Related Questions

Find the value of k for which the point (k-1, k) lies inside the parabola $$y^2$$ = 4x.

The focal distance of a point on the parabola $$y^2$$ = 12x is 4. Find the abscissa of this point.

The slope of the line touching both the parabolas $$y^2$$ = 4x and $$x^2$$ = -32 is