# Parabola Formula

A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola. Pascal stated that a parabola is a projection of a circle. Galileo explained that projectiles falling under the effect of uniform gravity follow a path called a parabolic path. Parabola Formula helps in representing the general form of this parabolic path in the plane.

In mathematics, any plane curve which is mirror-symmetrical and usually is of approximately U shared is called a parabola.

Let us study about parabola formula using solved examples.

## What Is a Parabola Formula?

The general equation of a parabola is:

**y = a(x - h) ^{2} + k (regular)**

**x = a(y - k) ^{2} + h (sideways)**

where,

(h,k) = vertex of the parabola

Four forms of a parabola and their formulas are:

**Example 1:** The equation of a parabola is y^{2 }= 24x. Find its length of latus rectum, focus, and vertex.

**Solution:**

To find: length of latus rectum, focus and vertex of a parabola

Given: equation of a parabola: y^{2 }= 24x

Therefore, 4a = 24

a = 24/4 = 6

Now, parabola formula for latus rectum:

Length of latus rectum = 4a

= 4(6)

= 24

Now, parabola formula for focus:

Focus = (a,0)

= (6,0)

Now, parabola formula for vertex:

Vertex = (0,0)

**Answer:** **Length of latus rectum = 24, focus = (6,0), vertex = (0,0)**

**Example 2: **The equation of a parabola is 2(y-3)^{2 }+ 24 = x. Find its length of latus rectum, focus, and vertex.

**Solution: **

To find: length of latus rectum, focus and vertex of a parabola

Given: equation of a parabola: 2(y-3)^{2 }+ 24 = x

On comparing it with the general equation of a parabola x = a(y-k)^{2 }+ h, we get

a = 2

(h, k) = (24, 4)

Now, parabola formula for latus rectum:

Length of latus rectum = 4a

= 4(2)

= 8

Now, parabola formula for focus:

Focus = (0,a)

= (0,2)

Now, parabola formula for vertex:

Vertex = (24,4)

**Answer:** **Length of latus rectum = 8, focus = (0, 2), vertex = (0, 0).**

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