Problem 22 - Intermediate Algebra

Question

Graph each equation. \(25 x^{2}+9 y^{2}=225\)

Step-by-Step Solution

Answer

The ellipse is centered at (0,0), has a vertical orientation with a semi-major axis of 5 and a semi-minor axis of 3, and it reaches from (-3,0) to (3,0) and (0,-5) to (0,5).

Step 1: Identify the Type of Conic Section

The given equation is \( 25x^2 + 9y^2 = 225 \). This is a form of the equation of an ellipse, which can be expressed as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \( a^2 \) and \( b^2 \) are denominators of the squared terms.

Step 2: Transform the Equation into Standard Form

To convert the equation into its standard form, divide every term by 225: \( \frac{25x^2}{225} + \frac{9y^2}{225} = 1 \). Simplifying the fractions, we get \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \). Hence, we have \( a^2 = 9 \) and \( b^2 = 25 \).

Step 3: Determine the Orientation of the Ellipse

In the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), if \( b^2 > a^2 \), the ellipse is vertically oriented. Here, \( b^2 = 25 \) and \( a^2 = 9 \), thus, the ellipse is vertical.

Step 4: Calculate the Semi-Major and Semi-Minor Axes

The semi-major axis \( b \) is the square root of \( b^2 = 25 \), hence \( b = 5 \). The semi-minor axis \( a \) is the square root of \( a^2 = 9 \), so \( a = 3 \).

Step 5: Identify the Center of the Ellipse

The standard form \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \) indicates that the center of the ellipse is at the origin (0, 0) because there are no \( x \) or \( y \) terms.

Step 6: Sketch the Ellipse

Using the semi-major axis 5 and semi-minor axis 3, draw the ellipse centered at the origin (0, 0). The ellipse reaches 5 units up and down from the center along the y-axis, and 3 units left and right along the x-axis.

Key Concepts

Conic Sections

When studying geometry, you'll come across conic sections, which are curves formed by slicing through a cone. These include circles, ellipses, parabolas, and hyperbolas. Each shape has unique properties depending on the angle and location of the slice.
Ellipses fall into this category and are particularly interesting. They are elongated circles, created when a plane cuts through a cone at an angle that is not perpendicular to the axis.

• They are defined formally as the set of all points where the sum of the distances from two foci is constant.
• You'll often encounter ellipses in various fields, from planetary orbits in astronomy to the physics governing sound and light.

Standard Form of Ellipse

Understanding the standard form of an ellipse is crucial for graphing and identifying its properties. In mathematics, an ellipse is often expressed in the equation:
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Here, the variables \(a^2\) and \(b^2\) are the squares of the semi-major and semi-minor axes, respectively.

• If \(a > b\), the ellipse stretches more along the x-axis, making it horizontally oriented.
• If \(b > a\), it stretches more along the y-axis, giving it a vertical orientation.

To utilize this form, you'll want to manipulate the ellipse equation by simplifying it, ensuring every term contributes to making the equation equal to 1. This step is key to easily identifying an ellipse’s dimensions and orientation.

Graphing Ellipses

Once you grasp the standard form, graphing an ellipse becomes straightforward. The process helps visually represent the ellipse's shape and key properties.

• First, determine the equation of the ellipse in standard form, separating the horizontal and vertical aspects.
• Identify the lengths of the semi-major and semi-minor axes by taking the square roots of \(a^2\) and \(b^2\).
• The ellipse is plotted by marking distances from the center along both the x-axis and y-axis based on the semi-axes' lengths.
• Sketch the curve ensuring a smooth, oval shape, equally elongating along the determined axes.

The center of the ellipse, often at the origin, plays an essential role. It grounds the symmetry of the shape as you draw. Ensure to label the axes lengths and center for complete understanding.

Vertical Orientation of Ellipse

Understanding the orientation of an ellipse is essential in graphing and applying the concept further. The orientation refers to whether the ellipse extends more along the x-axis or y-axis.
For an ellipse in its standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), a vertical orientation occurs when \(b^2 > a^2\).

• This taller shape extends more along the y-axis.
• Such ellipses are commonly seen in natural phenomena, such as the paths of certain celestial bodies or structures in architectural design.

Be sure to analyze the values of \(a\) and \(b\) carefully to determine orientation as it affects the overall placement and interpretation of the graph, providing crucial insight into the ellipse's spatial dynamics.