Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph. $$36 x^{2}+9 y^{2}+48 x-36 y+43=0$$

Step-by-Step Solution

Answer

The standard form of the ellipse equation is \(\left(x+ \frac{2}{3}\right)^2+\frac{1}{4}\left(y-2\right)^2 = 1\). The center is at \(- \frac{2}{3}, -2\), the vertices are at \(- \frac{2}{3}, - \frac{1}{2}\) and \(- \frac{2}{3}, - \frac{7}{2}\), the foci are at \(- \frac{2}{3}, - \frac{3}{2}\) and \(- \frac{2}{3}, - \frac{5}{2}\), and the eccentricity is \(\frac{2}{3} \).

Step 1: Write the Given Equation in the Standard Form

First, try to write the given ellipse equation in the standard form. For this, rearrange the equation and group the \(x\) terms together and the \(y\) terms together: \(36x^{2}+48x+9y^{2}-36y=-43\). Next, divide every term by 36 to make the coefficient of \(x^{2}\) and \(y^{2}\) as 1: \(x^{2}+ \frac{4}{3}x + \frac{1}{4}y^{2}- y = - \frac{43}{36}\). After that, complete the square for the \(x\) and \(y\) terms separately. This will give \(\left(x+ \frac{2}{3}\right)^2 - \left(\frac{2}{3}\right)^2 + \frac{1}{4} \left( y - \frac{2}{1}\right) ^ 2 - \left(\frac{2}{1}\right)^2 = - \frac{43}{36}\). Simplifying this will give the standard form of the ellipse.

Step 2: Find Properties of the Ellipse

Once you get the standard form of the ellipse, you can find the center, vertices, foci, and eccentricity of the ellipse. The center is given by the point \(-h, -k\) where \(h\) and \(k\) are the coefficients of \(x\) and \(y\) in the standard form. The vertices of the ellipse are given by \(\pm a\) along the major axis where \( a\) is the square root of the denominator of the second term. The foci are given by the points \(\pm c\) along the major axis where \(c = \sqrt{a^{2} - b^{2}}\). Finally, the eccentricity is given by \(e = \frac{c}{a}\).

Step 3: Sketching the Ellipse

Draw the coordinate axes and plot the center of the ellipse. Plot the vertices and foci along the major axis. Draw an ellipse shape that passes through these points. Alternatively, sketching can also be done using a graphing utility.

Key Concepts

Conic Sections

Conic sections are the curves obtained as the intersection of the surface of a cone with a plane. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. These shapes have unique properties and equations that define their structure.

An ellipse, which is the focus of our exercise, is formed when the plane cuts through the cone at an angle to its base, but does not intersect the base. It's characterized by its elongated shape, as opposed to a circle which is a special type of ellipse with equal radii.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic equation into a perfect square trinomial, thereby finding its simplified standard form. This method is pivotal in conic section analysis because it enables us to derive the standard equation of an ellipse from its general form.

To complete the square, you typically do the following steps: first, group the variable terms together on one side and the constant term on the other side. Then, for each variable, divide the coefficient of its linear term by 2, square the result, and add this square to both sides of the equation to maintain equality. The result is a transformed equation with a squared term and no linear term, revealing the center and other key features of the conic section.

Ellipse Properties

The standard form of an ellipse equation is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \(\left(h, k\right)\) is the center of the ellipse, and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. If \(a > b\), then the ellipse is stretched further along the x-axis, and vice versa.

The vertices of the ellipse are located \(a\) units from the center along the major axis, the foci are inside the ellipse on the major axis \(c\) units from the center where \(c = \sqrt{a^2 - b^2}\), and the eccentricity \(e\), which measures the 'ovalness' of the ellipse, is \(e = \frac{c}{a}\). Understanding these properties allows us to derive vital information about the ellipse's shape and position just from its equation.