Question

In the equation of an ellipse, \(4 x^{2}+9 y^{2}-16 x+18 y-11=0\) determine the standard form of the equation, and find the values of \(a, b, c\), and \(e\).

Short answer

The standard form of the given ellipse equation is \(\frac{(x - 2)^2}{9} + \frac{(y + 1)^2}{4} = 1\). The values for a, b, c, and e are: a = 3 b = 2 c = \(\sqrt{5}\) e =\(\frac{\sqrt{5}}{3}\)

Step-by-step solution

Rewrite the given equation

Let's rewrite the given equation as follows: \(4(x^2 - 4x) + 9(y^2 + 2y) = 11\)

Complete the square for x and y terms

Perform completing the square for both x and y terms: \(4(x^2 - 4x + 4) + 9(y^2 + 2y + 1) = 11 + 16 + 9\) So, we get the equation: \(4(x - 2)^2 + 9(y + 1)^2 = 36\)

Write the equation in standard form

Divide both sides of the equation by 36 to write the equation in its standard form: \(\frac{(x - 2)^2}{9} + \frac{(y + 1)^2}{4} = 1\)

Identify the values of a, b, c, and e

In this standard form, \(\frac{(x - 2)^2}{9} + \frac{(y + 1)^2}{4} = 1\), we are able to identify the values for a, b, h, and k. We have: 1. a^2 = 9, thus a = 3 2. b^2 = 4, thus b = 2 3. h = 2 4. k = -1 Since, c = \(\sqrt{a^2 - b^2}\), computing c we get: c =\(\sqrt{9 - 4}\) c = \(\sqrt{5}\) Now, we need to find the eccentricity e, which is given by: \(e =\frac{c}{a}\). Calculating e, we get: e =\(\frac{\sqrt{5}}{3}\)

Results

So, the standard form of the given ellipse equation is \(\frac{(x - 2)^2}{9} + \frac{(y + 1)^2}{4} = 1\). The values for a, b, c, and e are: a = 3 b = 2 c = \(\sqrt{5}\) e =\(\frac{\sqrt{5}}{3}\)

Key concepts

Completing the Square

When working with quadratic equations, such as those found in the equations of ellipses, 'completing the square' is a vital algebraic technique used to convert equations into a form that is easier to understand and work with. In the context of conic sections, this process helps in rewriting the equation into standard form, which is essential for identifying the properties of the ellipse.

Here's how completing the square works: Given a quadratic term like \(ax^2 + bx\), we add and subtract a certain value to create a perfect square trinomial \(ax^2 + bx + c\), where \(c = (b/2a)^2\). This new term \(c\) allows us to express the quadratic as \(a(x + d)^2\), where \(d = b/(2a)\). This format immediately gives us the center of the ellipse when applied to both \(x\) and \(y\) terms in its equation.

In our exercise, completing the square transforms the original equation into a clear and recognizable form that indicates the ellipse's center and axes lengths, key steps toward understanding its geometric properties.

Ellipse Eccentricity

Understanding the eccentricity of an ellipse is crucial as it describes the shape's 'roundness' or 'stretch.' It is a dimensionless number that quantifies how much an ellipse deviates from being a perfect circle. The eccentricity \(e\) of an ellipse can be found using the formula \(e = c/a\), where \(c\) is the distance from the center to a focus and \(a\) is the distance from the center to a vertex on the major axis.

An ellipse's eccentricity ranges from 0 to 1. An \(e = 0\) represents a perfect circle, as the foci and the center coincide, while an \(e \rightarrow 1\) indicates an increasingly elongated ellipse. In our exercise, once we've identified \(a\) and computed \(c\), we can determine that the eccentricity \(e = \frac{\sqrt{5}}{3}\) signifies that our ellipse is a bit more elongated than a circle but still less stretched out than an ellipse with an eccentricity closer to 1. The concept of eccentricity aids in visualization and comparative analysis of different ellipses.

Conic Sections

The term 'conic sections' encompasses the shapes formed by the intersection of a plane with a double-napped cone. These shapes include circles, ellipses, parabolas, and hyperbolas. The equation of an ellipse is one of the standard forms of conic sections, characterized by a set of points that the sum of the distances to two fixed points (foci) is constant.

In simpler terms, imagine slicing a cone with a flat surface. Depending on the angle and depth of the cut, you'll be able to produce different cross-sections, each representing a different conic section. Ellipses result from an oblique cut that goes through both nappes but does not pass through the apex of the cone.

This conic section's unique properties and equations make it a subject of study across multiple disciplines, from physics to astronomy. In our textbook problem, analyzing the equation of an ellipse \(4 x^2+9 y^2-16 x+18 y-11=0\) leads us to its standard form after applying the technique of completing the square, revealing the true nature of this conic section.