Eccentricity of Parabola

The eccentricity of parabola is the ratio of the distance of a point on the parabola from a fixed point (focus) and the perpendicular distance of the point from a fixed line (directrix). which is equal to one. The value of the eccentricity of a parabola is equal to one which can be obtained using the definition of a parabola. The formula for the eccentricity of a conic section is c/a, where c is the distance of the center from the focus and a is the distance of the center from the directrix. In the case of a parabola, we have c = a, and hence, the eccentricity of parabola becomes c/a which is equal to 1.

In this article, we will explore the concept of the eccentricity of parabola and its formula. We will prove its eccentricity and go through some solved examples based on the eccentricity of the parabola for a better understanding of the concept.

The eccentricity of parabola is equal to one. Let us recall the definition of a parabola and the eccentricity of a conic section. A parabola is a locus of a point that is equidistant from a fixed point, that is, focus and a fixed-line, that is, directrix. In simple words, we can say that any point on a parabola is equidistant from the focus and the directrix. On the other hand, the eccentricity of a conic section is the ratio of the distance of a point on the conic section from the focus to its perpendicular distance from the directrix.

For a parabola, the distance from a point on the parabola to the focus is equal to the perpendicular distance from the same point on the parabola to the directrix. Hence, the eccentricity of parabola is equal to one as the ratio becomes equal to 1.

The eccentricity of a parabola is defined as the mathematical constant which is given by the ratio of the distance of an arbitrary point P on the parabola from a fixed point - focus and the distance of the point P from corresponding the fixed-line - directrix. In a parabola, both these distances are equal for any arbitrary point P on the parabola. Hence, the ratio becomes one and the eccentricity of parabola is equal to 1.

The formula for the eccentricity of any conic section 'e' is equal to c/a, where c = distance of a point on the conic section from the focus and a = distance of the point on the conic section from the directrix. Assume an arbitrary point P on a parabola. Then, the formula for the eccentricity of parabola is given by,

Eccentricity = Distance of P From Focus / Distance of P From Directrix

⇒ e = c/a

Now that we know that the eccentricity of parabola is equal to one, let us derive this using the definition of a parabola. Consider a parabola with an arbitrary point P on it. Let F be the focus of the parabola and l be the directrix with a point M such that Pm is perpendicular to l on the directrix. According to the definition of a parabola, a parabola is a locus of the point P (as P is arbitrary) such that its distance from a fixed point F (Focus) is equal to its distance from a fixed line l (Directrix). Now, the eccentricity of a parabola is defined as the ratio of the distance of the arbitrary point P from the fixed point F and its distance from the fixed-line l.

Now, construct a perpendicular PM on the directrix l. Then, the eccentricity of parabola is given by e = PF/PM. Since the two distances are equal in the case of a parabola, therefore we have PF = PM. Hence, e = PF/PM = PF/PF = 1. Hence, we can say that the eccentricity of parabola is equal to one.

Important Notes on Eccentricity of Parabola

• The eccentricity of parabola is equal to one.
• A parabola is a locus of a point that is equidistant from a fixed point - focus and a fixed-line - directrix.
• The eccentricity of a conic section is the ratio of the distance of a point on the conic section from the focus to its perpendicular distance from the directrix.

☛ Related Topics:

• Eccentricity of Ellipse
• Eccentricity of Hyperbola
• Coordinate Geometry

What is the Eccentricity of Parabola?

The eccentricity of parabola is equal to one. It is the ratio of the distance of a point on the parabola from a fixed point (focus) and the perpendicular distance of the point from a fixed line (directrix).

How Do You Find the Eccentricity of Parabola?

The eccentricity of a parabola is defined as the mathematical constant which is given by the ratio of the distance of an arbitrary point P on the parabola from a fixed point - focus and the distance of the point P from corresponding the fixed-line - directrix. In the case of a parabola, we have these two distances as equal and hence, the eccentricity of the parabola is equal to 1.

What is the Formula for the Eccentricity of Parabola?

The formula for the eccentricity of parabola is given by,

Eccentricity = Distance of P From Focus / Distance of P From Directrix, where P is an arbitrary point on the parabola

⇒ e = c/a

What Does it Mean if Eccentricity is One?

If the eccentricity of a conic section is equal to one, then we can say that the conic section is nothing but a parabola because the eccentricity of parabola is equal to one.

Why Eccentricity of Parabola is 1?

The eccentricity of a parabola is equal to one as the distance of the center of a parabola from the focus is equal to the distance of the center of the parabola from the directrix for a parabola. We know that the eccentricity of a parabola is equal to the ratio of the distance of the center from the focus and its distance from the directrix. Therefore, the eccentricity of parabola is equal to one.