# Equation of a Parabola

Parabola can be defined as a type of conic section which forms the shape of a letter i.e. U. It is used in various topics of physics. One of the greatest scientists named Galileo explained the theory of the falling of projectiles and named it a parabolic path. It can also be considered as a curved shape object which is mirror-symmetrical. A parabola comprises various parts. Some of them are as follows, a focus, a focal distance and chord, a latus rectum, and many more. There are three types of equations given for a parabola. They are the standard equation, the regular and sideways equation. Amongst these three equations, the standard and regular are mostly used. They are written as, y = 4ax where x is the vertex of parabola and y = a (x – h) (x – h). We shall solve some examples related to the equation of the parabola in the coming sections.

**Equation of an Ellipse**

An ellipse is defined as the internal part of the conic section. It is often regarded as a circle due to its shape and properties. A circle is a type of geometrical shape which has no width and curves on it. An ellipse is regarded as the oval. The eccentricity of an ellipse is always measured as less than 1. The eccentricity of an ellipse can be defined as the ratio of the distance of the center and locus. There are two standard equations given for an ellipse. The first equation of an ellipse is written as x.x / b.b + y.y / a.a = 1 where x and y are the transverse and conjugate axis of the coordinate plane. As a parabola, an ellipse is also made with the help of various parts. Some of them are, a focus, major and minor axis, center, transverse and conjugate axis, and many more. The middlemost point which joins the two focuses of the ellipse is defined as a center. The length of the ellipse which is written as 2a can be considered as a major axis. Similarly, for 2b the part is defined as the minor axis.

**Examples Related to the Standard Equation of a Parabola**

The standard equation of a parabola is written as where x is the vertex of the parabola. Let us try to solve some examples related to the equation of parabola so that you can grasp the concept. Some of the examples are listed below:

Example 1: If the equation of Parabola is y.y = 32x, find the focus, vertex, and latus rectum of the parabola.

Solution**:** Given that,

Equation of parabola is given as = y.y = 32x.

We know that the standard equation given for a parabola is y = 4ax.

Thus, 4a = 32.

a = 8.

Now, to find the latus rectum of parabola,

4a = latus rectum.

4 * 8 = 32.

Then the focus of the parabola is = 8,0 and vertex = 0,0.

Therefore, the latus rectum, focus, and vertex of the parabola are equivalent to 32, (8, 0) and (0, 0).

Example 2: If the equation of Parabola is y.y = 40x, find the focus, vertex, and latus rectum of the parabola.

**Solution**: Given that,

Equation of parabola is given as = y.y = 40x.

We know that the standard equation given for a parabola is y = 4ax.

Thus, 4a = 40.

a = 10.

Now, to find the latus rectum of parabola,

4a = latus rectum.

4 * 10 = 40.

Then the focus of the parabola is = 10,0 and vertex = 0,0.

Therefore, the latus rectum, focus, and vertex of the parabola are equivalent to 40, (10, 0), and (0, 0).

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