Problem 25 - Algebra and Trigonometry

Question

Use a graphing device to graph the ellipse. $$ \frac{x^{2}}{25}+\frac{y^{2}}{20}=1 $$

Step-by-Step Solution

Answer

Graph the ellipse centered at the origin with vertices at \((\pm 5, 0)\) and co-vertices at \((0, \pm 2\sqrt{5})\).

Step 1: Identify the Equation Components

The given equation is \( \frac{x^2}{25} + \frac{y^2}{20} = 1 \). This is the standard form of an ellipse equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \( a^2 = 25 \) and \( b^2 = 20 \).

Step 2: Calculate the Values of 'a' and 'b'

To find the values of \( a \) and \( b \), calculate the square root of \( a^2 \) and \( b^2 \). Thus, \( a = \sqrt{25} = 5 \) and \( b = \sqrt{20} = 2\sqrt{5} \).

Step 3: Determine the Orientation

Since \( a^2 = 25 \) is greater than \( b^2 = 20 \), the major axis is along the x-axis. This tells us the ellipse is wider along the x-axis.

Step 4: Plot the Points on Major and Minor Axes

On the x-axis, plot the points at \( x = \pm a = \pm 5 \). On the y-axis, plot the points at \( y = \pm b = \pm 2\sqrt{5} \approx \pm 4.47 \). These points are the vertices and co-vertices of the ellipse.

Step 5: Graph the Ellipse

Using a graphing device, plot the ellipse by drawing an elongated circle passing through the plotted points, ensuring the curve is symmetric about both axes.

Key Concepts

Conic Sections

Conic sections are shapes created by intersecting a plane with a cone. These intersections produce different types of curves such as ellipses, circles, parabolas, and hyperbolas. Each curve is defined by its unique set of equations.

• Ellipses: Oval-shaped and defined by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) when centered at the origin.
• Circles: A special type of ellipse where \( a = b \), suggesting the curve has equal radii.
• Parabolas: Shaped like a 'U' and often appear in scenarios like projectile motion.
• Hyperbolas: Consisting of two disconnected curves facing away from each other.

Understanding how each conic section comes about helps in conceiving their geometric properties and applications.

Graphing Ellipses

Ellipses can be graphed by plotting their major and minor axes first and then shaping the curve around these points. The equation of the ellipse provides the information needed to plot these axes accurately.
For the equation \( \frac{x^2}{25} + \frac{y^2}{20} = 1 \):

• The major axis is determined by the larger denominator. In this instance, it's the x-axis since 25 is greater than 20.
• The minor axis is along the y-direction.
• The ellipse is wider along the x-axis, which signifies its horizontal orientation.

Use graphing software or manual sketching to plot the ellipse symmetrically about the center point. This guarantees the correct shape and scale representation.

Major and Minor Axes

The major and minor axes are crucial components of an ellipse. They determine the shape and orientation of the ellipse.

• Major Axis: The longest diameter of the ellipse, stretching across the widest part. Determined by the larger value of \( a^2 \) and \( b^2 \).
• Minor Axis: The shortest diameter of the ellipse, perpendicular to the major axis.

In our equation, \( a^2 = 25 \) and \( b^2 = 20 \), indicating that the major axis runs along the x-axis. The lengths of these axes are given by twice the values of 'a' and 'b':

• Length of major axis: \( 2 \times 5 = 10 \).
• Length of minor axis: \( 2 \times 2\sqrt{5} \approx 8.94 \).

These axes are pivotal for plotting an accurate curve of an ellipse.

Standard Form of Ellipse Equation

The standard form of an ellipse equation makes it easy to identify and plot ellipses. Given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the radii along the x- and y-axes, respectively.

• For a horizontally oriented ellipse: \( a > b \).
• For a vertically oriented ellipse: \( b > a \).

In our equation \( \frac{x^2}{25} + \frac{y^2}{20} = 1 \):

• It is already in standard form.
• The parameters indicate the major axis lies horizontally.

Recognizing this form in equations is the first step to understanding and graphing ellipses effectively, helping ensure the proper scale and orientation are applied.