Problem 18 - Thomas Calculus

Question

Exercises \(17-24\) give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. $$ 7 x^{2}+16 y^{2}=112 $$

Step-by-Step Solution

Answer

The standard form of the ellipse is \(\frac{x^2}{16} + \frac{y^2}{7} = 1\) with foci at \((3, 0)\) and \((-3, 0)\).

Step 1: Identify the Equation of Ellipse

We start with the equation of the ellipse given as \(7x^2 + 16y^2 = 112\). Our task is to convert this equation into the standard form of an ellipse.

Step 2: Divide Equation by Constant

To convert the equation into standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we need to divide the entire equation by 112.\[ \frac{7x^2}{112} + \frac{16y^2}{112} = \frac{112}{112} \]Simplifying gives us:\[ \frac{x^2}{16} + \frac{y^2}{7} = 1 \]

Step 3: Identify Parameters 'a' and 'b'

From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we identify \(a^2 = 16\) and \(b^2 = 7\). Thus, \(a = \sqrt{16} = 4\) and \(b = \sqrt{7}\). Since \(a > b\), it is a horizontal ellipse.

Step 4: Calculate the Foci

For an ellipse, the distance \(c\) of each focus from the center is given by \(c = \sqrt{a^2 - b^2}\).\[ c = \sqrt{16 - 7} = \sqrt{9} = 3 \]The foci are located at \((\pm c, 0)\) in a horizontal ellipse. Thus, the foci are \((3, 0)\) and \((-3, 0)\).

Step 5: Sketch the Ellipse

Now, sketch the ellipse based on the identified parameters:- The center is at the origin \((0, 0)\).- The major axis is horizontal with length \(2a = 8\), extending from \((-4, 0)\) to \((4, 0)\).- The minor axis is vertical with length \(2b = 2\sqrt{7}\), extending from \((0, -\sqrt{7})\) to \((0, \sqrt{7})\).- Mark the foci at \((3, 0)\) and \((-3, 0)\).

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse's equation is crucial to understanding the shape and position of the ellipse on a coordinate plane. You'll often see it written as:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

In this formula, \(a^2\) and \(b^2\) are the denominators under the squared terms, and they relate directly to the lengths of the ellipse's axes. When transforming any given ellipse equation into this standard form, the goal is to shape it into this template.

For instance, if presented with an equation like \(7x^2 + 16y^2 = 112\), you'd start by dividing every term by 112 to ensure the equation equals 1. This manipulation is essential because the standard form requires that the sum of the terms on the left be equal to 1.

Once you've adjusted the equation,\(\frac{x^2}{16} + \frac{y^2}{7} = 1\) emerges as the standard form. Here, it's clear that \(a^2 = 16\) and \(b^2 = 7\), which determine the ellipse's dimensions.

Foci of an Ellipse

The foci of an ellipse are two special points located along the major axis. These points are key to the geometric definition of an ellipse, which involves the constant distance from any point on the ellipse perimeter to the foci. Understanding the foci is like grasping the heart of an ellipse’s symmetry and shape.

To find the foci, we use the formula:\[ c = \sqrt{a^2 - b^2} \]
Here, \(c\) represents the distance from the ellipse's center to each focus. This formula requires that \(a > b\), ensuring we correctly identify the major axis. Placements are then determined along this axis.

In our current example, since \(a^2 = 16\) and \(b^2 = 7\), solving gives \(c = \sqrt{16 - 7} = 3\). Thus, for a horizontal ellipse centered at the origin, the foci will be at the points \((3, 0)\) and \((-3, 0)\). Knowing where these foci are helps in drawing a more accurate ellipse and provides deeper insight into its properties.

Major and Minor Axes

These terms describe the longest and shortest diameters of an ellipse, critical for understanding the ellipse's orientation and size. The "major axis" is always the longer one, spanning from one end of the ellipse to the other, passing through the center and both foci.

The "minor axis" sits perpendicular to the major axis at the center, providing the ellipse’s width. In equations, \(a\) and \(b\) are the semi-major and semi-minor axes, respectively; thus, the axes' lengths are \(2a\) and \(2b\).

In our example, the major axis is horizontal, with \(a = 4\); hence, the full major axis measures \(2a = 8\) and spans from \((-4, 0)\) to \((4, 0)\). For the minor axis, \(b = \sqrt{7}\), resulting in a total length of \(2b = 2\sqrt{7}\), extending vertically from \((0, -\sqrt{7})\) to \((0, \sqrt{7})\). Understanding these axes make it easier to visualize the ellipse's shape and helps in sketching.