Problem 11 - College Algebra

Question

Graph each ellipse and locate the foci. $$ x^{2}=1-4 y^{2} $$

Step-by-Step Solution

Answer

The ellipse is centered at the origin, has its major axis along the x-axis with a length of 2a (or 2), its minor axis is of length 2b (or 1). The foci of the ellipse are located at the points \((\sqrt{3}/2, 0)\) and \(-(\sqrt{3}/2, 0)\).

Step 1: Write the Expression in Standard Form

Rewrite the equation \(x^{2}=1-4y^{2}\) in standard form of an ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Divide both sides by \(1\), resulting in \(x^{2} + 4y^{2} = 1\) and then rearrange to put in standard form \(\frac{x^{2}}{1} + \frac{y^{2}}{1/4} = 1\). Here, \(a=1\) and \(b=1/2\).

Step 2: Plot the Ellipse

Since \(a^{2} = 1\) and \(b^{2} = 1/4\), we have \(a = 1\) and \(b = 0.5\) (major axis along x-axis since \(a > b\)). The center is at the origin (0,0). Plot the points (1,0), (-1,0), (0,0.5), and (0,-0.5). Draw the ellipse that goes through these points.

Step 3: Locate the Foci

The distance from the center to the foci, c, is given by the equation \(c = \sqrt{a^{2}-b^{2}}\). In this case, \(c = \sqrt{1 - (1/4)} = \sqrt{3/4} = \sqrt{3}/2\). Therefore, the foci are located at \((\sqrt{3}/2, 0)\) and \(-(\sqrt{3}/2, 0)\).

Key Concepts

Standard Form of an Ellipse

Understanding the standard form of an ellipse is crucial for graphing and analyzing its properties. The standard form equation for an ellipse with a center at the origin (0,0) is written as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) denote the lengths of the semi-major and semi-minor axes, respectively. If the ellipse is horizontal, \(a\) is associated with the x-axis, and if it's vertical, \(a\) is on the y-axis. If \(a > b\), the ellipse extends further along the x-axis, and if \(b > a\), it stretches more along the y-axis.

Keeping this equation in mind, we can transform any quadratic equation into this standard form to determine the specific characteristics of the ellipse, such as its orientation and dimensions. For instance, taking the example \(x^2 = 1 - 4y^2\) and rearranging it into standard form allows us to clearly see that it's a horizontal ellipse with a semi-major axis of 1 and a semi-minor axis of 0.5. By visualizing this, graphing becomes much more manageable.

Ellipse Foci

The foci (plural of focus) of an ellipse are two points located symmetrically on the major axis of the ellipse. They are one of the defining features of an ellipse as the sum of the distances from the foci to any point on the ellipse is constant. The foci play a vital role in various physics and engineering problems, such as in the design of elliptical reflectors and orbits of planets.

To locate the foci on a standard form ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^{2}} = 1\), you use the distance formula \(c = \sqrt{a^2 - b^2}\), where \(c\) is the distance from the center to either focus. For our example, with \(a = 1\) and \(b = 0.5\), computing \(c\) gives us \(\sqrt{3}/2\), which means the foci are located at \(\pm(\sqrt{3}/2, 0)\) along the x-axis. Understanding how to find the foci is vital, as these points can determine the shape of the ellipse and are often necessary for solving real-world problems involving elliptical paths.

Conic Sections Algebra

Conic sections are the curves obtained by intersecting a plane with a double-napped right circular cone. These include ellipses, circles (a special case of ellipses), parabolas, and hyperbolas. Algebraically, we can describe conic sections by second-degree polynomial equations in two variables. Understanding the algebra behind these conic sections enables students to solve complex problems by simply manipulating equations.

For example, the original equation \(x^2 = 1 - 4y^2\) can be transformed into the standard form of an ellipse by completing the square or equivalent algebraic manipulations. This transformation highlights one core algebraic skill necessary for working with conics: the ability to recognize and rewrite quadratic equations into their canonical forms to reveal the underlying geometric figure. By mastering conic sections algebra, students can tackle a broad range of problems, from graphing simple curves to understanding the orbits of celestial bodies.