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Chapter 10: Problem 17

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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 \((0, \sqrt{2})\).
04

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 \((0, \sqrt{2})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. You told me that an ellipse centered at the origin has vertices at \((-5,0)\) and \((5,0),\) so 1 was able to graph the ellipse.

In Exercises \(61-66,\) find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{array}{r} {\frac{x^{2}}{25}+\frac{y^{2}}{9}=1} \\ {y=3} \end{array}\right. $$

Use a graphing utility to graph the parabolas in Exercises 86鈥�87. Write the given equation as a quadratic equation in y and use the quadratic formula to solve for y. Enter each of the equations to produce the complete graph. $$ y^{2}+2 y-6 x+13=0 $$

Describe how to graph \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)

Solve by eliminating variables: $$\left\\{\begin{aligned} x-6 y &=-22 \\ 2 x+4 y-3 z &=29 \\ 3 x-2 y+5 z &=-17 \end{aligned}\right.$$
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