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Chapter 10: Problem 16

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ 4 x^{2}+25 y^{2}=100 $$

Short Answer

Expert verified
The ellipse described by the equation \(4x^2 + 25y^2 = 100\) can be graphed with semi-major axis 5 units along the x-axis, semi-minor axis 2 units along the y-axis, and foci at approximately (卤4.58, 0).

Step by step solution

01

Rewrite the Equation in Standard Form

The first task involves rewriting the given equation, \(4x^2 + 25y^2 = 100\), in standard form. This can be achieved by dividing through by 100, yielding \(\frac{x^2}{25} + \frac{y^2}{4} = 1\).
02

Identify the Values of \(a\) and \(b\)

Next, identify the values of \(a^2\) and \(b^2\) as given by the standard form, such that \(a^2 = 25\) which makes \(a = 5\) and \(b^2 = 4\) which makes \(b = 2\).
03

Determine the Coordinates of the Foci

The foci are located at \(c = \sqrt{a^2 - b^2}\). In this case, this comes to \(c = \sqrt{25 - 4} = \sqrt{21}\), or approximately 卤4.58 units on the x-axis from the center of the ellipse (which is the origin, since there are no h or k values in the equation). Thus, the coordinates of the foci are approximately (卤4.58, 0).
04

Graph the Ellipse

Finally, draw the ellipse by plotting the points (卤5, 0) and (0, 卤2) for the semi-major and semi-minor axes, respectively, and sketching a smooth curve. Then plot the foci at (卤4.58, 0).

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Key Concepts

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

Standard Form of an Ellipse
When working with ellipses, the standard form of an equation is essential. This form allows us to easily identify and graph the ellipse. The standard equation for an ellipse centered at the origin is given by:
  • For a horizontal ellipse: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
  • For a vertical ellipse: \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]
Where \(a\) represents the length of the semi-major axis and \(b\) represents the length of the semi-minor axis. Converting a general form equation such as \(4x^2 + 25y^2 = 100\) to the standard form involves dividing every term by 100 to achieve \[ \frac{x^2}{25} + \frac{y^2}{4} = 1 \]. This reveals that it's a horizontal ellipse centered at the origin. Whenever you work with ellipses, always aim to convert to this standard form.
Foci of an Ellipse
The foci are two special points inside an ellipse that define its shape. They're crucial for understanding the ellipse's geometry and are used for constructing the curve itself. For an ellipse centered at the origin, the foci lie along the major axis. The distance of the foci from the center is determined by: \[ c = \sqrt{a^2 - b^2} \] In an example where \(a = 5\) and \(b = 2\), the foci distance \(c\) is calculated as \(\sqrt{25 - 4} = \sqrt{21}\). Thus, the foci are located approximately at (卤4.58, 0) on the x-axis. Always remember, the foci lie closer to the center than any of the axes' endpoints.
Semi-Major and Semi-Minor Axes
Understanding the semi-major and semi-minor axes helps in graphing the ellipse accurately. The semi-major axis is the longest radius of the ellipse, while the semi-minor axis is the shortest.
  • Semi-Major Axis: Denoted as \(a\), it's the longest distance from the center to the edge of the ellipse. In the equation \(\frac{x^2}{25} + \frac{y^2}{4} = 1\), it is determined to be 5 units along the x-axis (卤5, 0).
  • Semi-Minor Axis: Denoted as \(b\), it represents the shortest distance from the center to the edge. It measures 2 units along the y-axis (0, 卤2).
These axes determine the overall shape and direction of the ellipse on a graph. Draw them first to serve as guidelines when sketching the ellipse.
Ellipse Equation Conversion
Converting between different forms of an ellipse equation is critical for analyzing and graphing ellipses. Starting with a general form, such as \(4x^2 + 25y^2 = 100\), the goal is to achieve the neat standard form \(\frac{x^2}{25} + \frac{y^2}{4} = 1\). To accomplish this:
  • Identify the constants with \(x^2\) and \(y^2\) terms. Here, these are 4 and 25.
  • Divide each term of the equation by the number equal to the sum on the right side, which in this example is 100.
  • Simplify to reveal the fractions, placing the equation in the form of \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
Through this conversion, we gain valuable and immediate visual insights into the ellipse's size and orientation, enabling precise graphing.

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