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.

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.

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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