Eccentricity of Parabola

Eccentricity of Parabola is 1.

Eccentricity of a parabola is a measure of its deviation from a perfect circle. It’s a key parameter that describes the shape and behavior of the parabolic curve. Unlike ellipses and hyperbolas, which have eccentricities greater than or equal to 1, a parabola has an eccentricity exactly equal to 1. In this article, we will discuss the eccentricity of a parabola in detail, including it’s value as well as its derivation.

What is Parabola?

A parabola is a type of curve in mathematics that is commonly seen in algebra and geometry. It’s a symmetrical curve that can either open upwards or downwards.

Parabolas are seen in various real-world phenomena, including projectile motion (like the path of a thrown object), the shape of satellite dishes, the trajectory of celestial bodies under gravity, and the reflective properties of certain surfaces, like mirrors and antennas.

General Equation of a Parabola

The general equation of a parabola is:

x² = 4ay

Where a is the distance from the vertex to the focus.

This equation can be derived using the definition of a parabola and the properties of similar triangles. A parabola’s eccentricity is always 1, regardless of the value of a.

Examples of Parabola

The examples of the eccentricity of Parabola are:

• Satellite Dishes: Parabolic reflectors with an eccentricity of 1 are commonly used in satellite dishes to collect and focus incoming signals to a single point efficiently.
• Reflecting Telescopes: The mirrors in reflecting telescopes are often shaped as parabolic reflectors with an eccentricity of 1 to gather and concentrate light rays for more precise imaging.
• Parabolic Arch Bridges: The design of parabolic arch bridges utilizes the strength and stability provided by the shape of a parabola with an eccentricity of 1 to distribute weight and stress evenly.
• Solar Cookers: Parabolic reflectors with an eccentricity of 1 are employed in solar cookers to concentrate sunlight onto a focal point for cooking or heating purposes.
• Headlights: Some high-end car headlights use parabolic reflectors with an eccentricity of 1 to focus and effectively direct light beams for better night visibility.

What Is Eccentricity of Parabola?

The word “eccentricity” measures how near to circularity or how far from it the curvature of a curve gets. Eccentricity, for the parabola, remains always 1.

The parabola is a curve with points equidistant from a focus point and from a line called the directrix. From this, we learn that the length from any point on the focus to the focal point is equal to the distance perpendicular to any point on the parabola and the directrix.

Accordingly, these distances have a ratio always equal to 1, and, as a result, the eccentricity of the parabola is 1. Studying eccentricity can help us determine a parabola’s focus and directrix, and these concepts govern various practical matters.

Formula of Eccentricity of Parabola

The formula for the eccentricity of a parabola is:

e = c/a

Here, e stands for the eccentricity, c is the distance of the parabola’s point from its focus, and a is the distance between the parabola’s point and the directrix.

This formula stresses the relationship between the distances from points on the parabola to its focus and directrix, making these distances equal. Hence, the value of its eccentricity is the constant number 1.

Derivation of Eccentricity of Parabola

Consider a parabola M with any point P on it. Let F be the parabola’s focus and l be the directrix with a point from them where Pm is perpendicular to l on the directrix.

Parabola can be defined geometrically as the path in which the point P (as P is an arbitrary figure) follows, and its distance from a fixed point F (Focus) is equal to its distance from the directrix l.

Therefore, we have,

PF = PM

By definition, the eccentricity of a parabola that touches an arbitrary point P from the fixed point F and which is simultaneously perpendicular to the directrix is the ratio of the distance from point P to point F and the perpendicular distance of point P to the directrix.

Hence,

e = PF/PM

Since the two distances are equal in the case of a parabola, we have:

e = PF/PM = PF/PF = 1

Therefore, the eccentricity of a parabola is equal to one.

Conclusion

The eccentricity of a parabola is a fundamental property that defines its shape and behavior. As we have explored, the eccentricity of a parabola is always equal to 1, which means that a parabola is a conic section with a constant eccentricity.

This unique property of parabolas has significant implications in various fields, from physics and engineering to architecture and art. The constant eccentricity of a parabola allows for the efficient design of structures, the accurate prediction of the motion of projectiles, and the creation of aesthetically pleasing curves in design.

Read More,

• Conic Sections
• Eccentricity Formula
• Parabola
• Vertex of a Parabola Formula
• Standard Equation of a Parabola
• Focus and Directrix of a Parabola

Solved Questions on Eccentricity of Parabola

Question 1: Find the vertex, focus, and directrix of the parabola y = 1/2 x².

Answer:

The given equation of the parabola is y = 1/2 x².

To find the vertex, we can rewrite the equation in vertex form:

y = a(x – h)² + k

Where (h, k) is the vertex.

In this case, a = 1/2 and h = 0, k = 0.So, the vertex of the parabola is (0, 0).

To find the focus, we use the formula:

Focus: (h, k + 1/(4a)) = (0, 1/4)

To find the directrix, we use the formula:

Directrix: y = k – 1/(4a) = -1/4

The parabola graph is a U-shaped curve opening upwards, with the vertex at (0, 0), the focus at (0, 1/4), and the directrix at y = -1/4.

Question 2: Find the standard form of the parabola equation with its vertex at the origin and focus at (0, -3/2).

Answer:

Given that the vertex is at the origin (0, 0) and the focus is at (0, -3/2).

We can use the formula for the standard form of the equation of a parabola with its vertex at the origin:

y = ax²

Since the focus is below the x-axis, we know that a is negative.

To find the value of a, we use the formula:

c = -1/(4a)

Where c is the distance from the vertex to the focus.

In this case, c = -3/2.So, we have:-3/2 = -1/(4a)

Solving for a, we get:

a = -1/6

Therefore, the standard form of the equation of the parabola is:

y = -1/6 x²

Practice Problems on Eccentricity of Parabola

Problem 1: Find the eccentricity of a parabola with a focus at (3, 0) and a directrix given by the equation x = −3.

Problem 2: Determine the eccentricity of the parabola defined by the equation y² = 4px where p = 3.

Problem 3: Given the equation of a parabola as x²=−16y, calculate its eccentricity.

Problem 4: A parabolic mirror has a focal length of 8 cm. Determine the eccentricity of this parabola.

Problem 5: Find the eccentricity of the parabola defined by the equation x² − 6x + 5.

FAQs on Eccentricity of Parabola

What is the eccentricity of a parabola?

The eccentricity of a parabola is a measure of its deviation from a perfect circle.

How is the eccentricity of a parabola defined?

The eccentricity of a parabola is defined as the ratio of the distance between a point on the parabola and its focus to the distance between the point and its directrix.

What is the value of eccentricity of a parabola?

Eccentricity of a parabola is always equal to 1.

What is the eccentricity of a parabola compared to other conic sections?

Unlike ellipses and hyperbolas, which have eccentricities greater than or equal to 1, a parabola has an eccentricity exactly equal to 1.

What if the eccentricity is zero?

Given that the eccentricity is zero, the parabola changes into a line.

What is the symbol for eccentricity?

The symbol for “eccentricity” is usually given by the letter “e.”

How is the eccentricity of a parabola calculated?

The eccentricity of a parabola can be calculated using its equation. For a standard vertical parabola of the form y = ax² + bx + c, the eccentricity e is given by e = .