Problem 13 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The length of the minor axis (2a) is 4, the length of the major axis (2b) is 10, and the foci are at \([- \sqrt{21}, 0]\) and \([+\sqrt{21}, 0]\). The ellipse is elongated in the y-direction.

Step 1: Express the equation in the standard form

First, put the given equation, \(25x^2+4y^2=100\), in the standard form of an ellipse, which is \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). This can be done by dividing everything by 100. We get: \(\frac{x^2}{4}+\frac{y^2}{25}=1\).

Step 2: Determine the major and minor axes

In this form, the number under \(x^2\) is \(a^2\) and under \(y^2\) is \(b^2\), so \(a^2=4\) and \(b^2=25\). The length of the major axis is 2b, so 2*5 = 10. The length of the minor axis is 2a, so 2*2 = 4.

Step 3: Calculate the distance of each focus from the center

We then calculate the distances of the foci from the center using the formula \(c=\sqrt{a^2-b^2} = \sqrt{25-4} = \sqrt{21}\). So, the foci are at \([- \sqrt{21}, 0]\) and \([+\sqrt{21}, 0]\).

Step 4: Plotting the ellipse

The ellipse can now be drawn using the obtained lengths of the axes and the positions of the foci. It will be elongated in the y-direction because b > a.

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse is a clear method to express the geometry of this shape and is crucial for both graphing and analyzing its properties. To bring an ellipse equation into its standard form, it's typically written as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) for an ellipse centered at the origin, where \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively.

In the worked example, the given equation \(25x^2 + 4y^2 = 100\) is rewritten in standard form by dividing each term by 100, resulting in \(\frac{x^2}{4} + \frac{y^2}{25} = 1\). This form is easier to interpret and analyze as it reveals the orientation and relative lengths of the axes directly, facilitating the graphing process and comprehension of the ellipse's shape.

Major and Minor Axes of an Ellipse

The major and minor axes are two special diameters of an ellipse which help define its shape and orientation. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest.

Length of Major and Minor Axes

For an ellipse in standard form, the length of the semi-major axis \(a\) and the semi-minor axis \(b\) can be extracted directly from the equation. They are represented as follows: \(a = \sqrt{a^2}\) and \(b = \sqrt{b^2}\).

In our example, the length of the semi-major axis is \(b = \sqrt{25} = 5\) and of the semi-minor axis is \(a = \sqrt{4} = 2\). Therefore, the lengths of the full major and minor axes are \(2b = 10\) and \(2a = 4\), respectively. This understanding is pivotal as it provides a visual framework for plotting the ellipse.

Foci of an Ellipse

The foci (singular: focus) of an ellipse are two specific points located along the major axis, equidistant from the center, and they play a crucial role in the formal definition of an ellipse. The sum of the distances from any point on the ellipse to the two foci is constant and equals the length of the major axis.

Calculating the Foci

For an ellipse in standard form, the distance \(c\) from the center to each focus can be found using the formula \(c = \sqrt{a^2 - b^2}\) if \(a\geq b\), which is the case for ellipses elongated along the y-axis. Conversely, for ellipses elongated along the x-axis, we use \(c = \sqrt{b^2 - a^2}\).

In the given example, where \(b > a\), the foci are computed using \(c = \sqrt{25 - 4} = \sqrt{21}\). The foci are, therefore, points at which \(x = \pm\sqrt{21}\) and \(y = 0\), confirming the ellipse's orientation along the y-axis.

Ellipse Equation

The equation of an ellipse provides a mathematical representation that describes the set of all points \( (x, y) \) that form the shape. An ellipse with a center at the origin, horizontal major axis, and vertical minor axis would have the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a > b\). Conversely, if the major axis is vertical, as observed in the exercise, the equation is written with \(b > a\).

Understanding the Equation

The coordinate pairs that satisfy this equation lie on the ellipse. Graphing becomes straightforward by plotting key points and the foci, then sketching the curve that encompasses these loci. For the provided example, using the standard form and calculated values for \(a\) and \(b\), and knowing the foci, we can plot the ellipse by drawing a rectangle with sides parallel to the axes, with lengths equal to the major and minor axes. The ellipse will fit neatly within this rectangle, touching each midpoint of the rectangle's sides.

The equation is the footprint guiding us through the shape's construction and is always the starting point for graphing an ellipse.