Problem 17 - Algebra and Trigonometry

Question

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ 7 x^{2}=35-5 y^{2} $$

Step-by-Step Solution

Answer

The graph of the given ellipse equation is vertically oriented with its center at the origin. The foci are located at \((0, -\sqrt{2}\)) and \((0, \sqrt{2})\).

Step 1: Rewrite the Equation in Standard Form

First, rewrite the given equation \(7x^2 = 35 - 5y^2\) by dividing both sides by 35. This gives \(x^2/5 + y^2/7 = 1\), which is the standard form of an ellipse.

Step 2: Identify the Values of a and b

The standard form of an ellipse is \(x^2/a^2 + y^2/b^2 = 1\). In our case, \(a^2 = 5\) and \(b^2 = 7\), hence \(a = \sqrt{5}\) and \(b = \sqrt{7}\). The value of \(a\) is less than \(b\), which means this is a vertically oriented ellipse.

Step 3: Locate the Foci

For an ellipse, the distance from the center to the foci is given by \(c = \sqrt{b^2 - a^2}\). Here, \(c = \sqrt{7 - 5} = \sqrt{2}\), so the foci are at \((0, -\sqrt{2})\) and \((0, \sqrt{2})\).

Step 4: Sketch the Ellipse

Now draw the ellipse with the information above. The center is at the origin (0,0), the major axis has length \(2b = 2\sqrt{7}\) along the y-axis, and the minor axis has length \(2a = 2\sqrt{5}\) along the x-axis. The foci are at \((0, -\sqrt{2})\) and \((0, \sqrt{2})\).

Key Concepts

Ellipse Equation

An ellipse is a fascinating geometric shape that you often encounter in mathematics, especially in algebra and geometry. The general form of an ellipse equation is \(Ax^2 + By^2 = C\). To make an ellipse equation more manipulable, it is common to rewrite it in terms of fractions so it resembles the standard form. This formatting makes identifying the properties of the ellipse straightforward.By converting \(7x^2 + 5y^2 = 35\) to \(x^2/5 + y^2/7 = 1\), you now possess an equation where variables are isolated with unit terms on the right-hand side. This form sets the stage for further exploration into the ellipse’s dimensions and orientation.

Foci of an Ellipse

Within an ellipse, points called "foci" hold a unique property. The sum of the distances from any point on the ellipse to each of the foci remains constant. This fundamental characteristic defines the shape of the ellipse.For our ellipse \(x^2/5 + y^2/7 = 1\), you find the foci using the formula \(c = \sqrt{b^2 - a^2}\), where \(b\\) and \(a\\) define the semi-major and semi-minor axes.In this case, \(a = \sqrt{5}\) and \(b = \sqrt{7}\) lead to \(c = \sqrt{7 - 5} = \sqrt{2}\). The foci are positioned on the y-axis at \((0, \sqrt{2})\) and \((0, -\sqrt{2})\). These locations play a critical role in understanding the nature of the ellipse.

Standard Form of an Ellipse

The standard form of an ellipse equation is crucial for quickly discerning an ellipse's properties. It is structured as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) for an ellipse centered at the origin with axes aligned to the coordinate planes.When you rewrite an equation into this form, it provides direct insights:

• The terms \(a\\) and \(b\\) represent the lengths of the semi-minor and semi-major axes.
• The structure dictates whether the ellipse is vertically or horizontally oriented.

For our equation, \(x^2/5 + y^2/7 = 1\), since \(b > a\), the ellipse is vertically oriented, wrapping itself around the y-axis. Placing the equation this way streamlines further analysis, like finding the foci and sketching the shape.

Major and Minor Axes

An ellipse consists of two main axes: the major and the minor axes. These axes give the ellipse its characteristic stretched or elongated appearance.For our example, since \(b > a\), the major axis is aligned with the y-axis and measures \(2b = 2\sqrt{7}\) in length. This is the axis along which the ellipse is longest. The minor axis, shorter in comparison, is along the x-axis and measures \(2a = 2\sqrt{5}\) in length. By understanding these axes, you can more precisely sketch the ellipse and gain insights into its geometric properties.