Problem 71 - Precalculus with Limits

Question

In Exercises 69-72, sketch the graph of the ellipse, using latera recta (see Exercise 68). \(5x^2+3y^2=15\)

Step-by-Step Solution

Answer

The ellipse centered at the origin has its major axis along the y-axis with endpoints at (0, sqrt(5)) and (0, -sqrt(5)) and minor axis along the x-axis with endpoints at (sqrt(3), 0) and (-sqrt(3), 0).

Step 1: Convert to Standard Form

Divide the entire equation by the constant on the right side to get it in standard form, which is \((x^2/a^2) + (y^2/b^2) = 1\). When we divide each term of the equation \(5x^2 + 3y^2 = 15\) by 15, we get \((x^2/3) + (y^2/5) = 1\). Here, a^2 = 3 and b^2 = 5.

Step 2: Identify the lengths of the major and minor axis

The square root of a^2 and b^2 will give the lengths of the semi-major and semi-minor axis respectively. So, a = sqrt(3) and b = sqrt(5). Since b>a, the major axis is along the y-axis and the minor axis is along the x-axis. A point on the major axis and a point on the minor axis will provide the latera recta.

Step 3: Sketch the Ellipse

Draw the x and y axis on the graph paper. Mark the x-values going from -sqrt(3) to sqrt(3) and y-values going from -sqrt(5) to sqrt(5). Draw the ellipse using these marked points. The end points of the major axis are at (0, sqrt(5)) and (0, -sqrt(5)) and the end points of the minor axis are at (sqrt(3), 0) and (-sqrt(3), 0). Finally, add the latera recta, which are lines passing through the foci parallel to the minor axis. In this case, since the ellipse is centered at the origin and is vertically elongated, the foci will lie along the y-axis, at a distance of sqrt(|b^2 - a^2|) = sqrt(2) from the origin.

Key Concepts

Standard Form of Ellipse Equation

When graphing an ellipse, the first step is to recognize the standard form of its equation. In trigonometry and coordinate geometry, the standard form is a blueprint that guides us to all pertinent information about the ellipse. It is written as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) for horizontal ellipses, or \(\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1\) for vertical ellipses, where 'a' and 'b' are the lengths of the semi-major and semi-minor axes respectively.

To convert a given equation to standard form, like the example exercise given with \(5x^2 + 3y^2 = 15\), we divide by 15 to get \(\frac{x^2}{3} + \frac{y^2}{5} = 1\). This aligns with the standard form where \(a^2 = 3\) and \(b^2 = 5\) after the division. Understanding this transformation is crucial as it reveals the ellipse's orientation and the relative sizes of the axes.

Major and Minor Axes of an Ellipse

Ellipses have two distinct axes, the major and minor axes, which are crucial to their structure. The longest diameter is known as the major axis, while the shortest is called the minor axis. For our example, the semi-major axis is sqrt(5) and the semi-minor axis is sqrt(3), as derived from the standard form equation.

The orientation of the ellipse is determined by which axis is longer; in this case, since \(b^2 = 5\) is greater than \(a^2 = 3\), the major axis lies along the y-axis, and the minor axis along the x-axis. Consistently recognizing these axes will aid in accurate plotting of the ellipse.

Latera Recta of an Ellipse

A lesser-known but equally fascinating feature of an ellipse is the 'latera recta'. These are lines that pass through the foci of the ellipse and are parallel to the minor axis.

The distance of the latera recta from the center can be determined by calculating \(\sqrt{|b^2 - a^2|}\). For our ellipse, this gives us \(\sqrt{2}\) as the foci lie along the y-axis (since it's the major axis). The latera recta, although not as commonly used as the axes, provide more insight into the ellipse's geometry and can serve as an additional guide during sketching.

Sketching Ellipses

To sketch an ellipse, start by drawing the coordinate axes. Mark the lengths of the semi-major and semi-minor axes on their respective lines using the values determined from the standard form equation. For our equation, we plot the points at \(\pm\sqrt{3}\) along the x-axis and \(\pm\sqrt{5}\) along the y-axis.

Once you've marked the axes, draw a smooth curve to connect these points, ensuring the curve is symmetrical about both axes. The intersection points of the axes and the curve represent the vertices of the ellipse. Adding the latera recta, which in this scenario would be lines parallel to the x-axis at \(\pm\sqrt{2}\) distances from the center, helps finalize the outline of the ellipse. A properly sketched ellipse should encapsulate all these elements, creating an accurate representation of the given equation.