Problem 3 - College Algebra

Question

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$\frac{x^{2}}{9}+\frac{y^{2}}{36}=1$$

Step-by-Step Solution

Answer

The semi-major axis \(a=3\) and the semi-minor axis \(b=6\). The ellipse is plotted with the center at the origin, the major axis along the y-axis from \(-6\) to \(6\) and the minor axis along the x-axis from \(-3\) to \(3\). The foci are \((0,5)\) and \((0,-5)\).

Step 1: Identify the semi-major axis and semi-minor axis

From the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{36}=1\), we can see that \(a^{2}=9\) and \(b^{2}=36\). So, the semi-major axis \(a=3\) and the semi-minor axis \(b=6\).

Step 2: Graph the ellipse

Plot the center of the ellipse at the origin point \((0,0)\). The semi-major axis extends along the y-axis from \(0\) to \(6\) and from \(0\) to \(-6\). The semi-minor axis extends along the x-axis from \(0\) to \(3\) and from \(0\) to \(-3\). Plot these points and draw the ellipse.

Step 3: Calculate foci points

To calculate the foci, use the formula \(c=\sqrt{b^2-a^2}\) where \(c\) is the distance from the center to each of the foci. Substituting the given values, we get \(c=\sqrt{36-9}=5\). The foci are located at \(0\) plus or minus \(c\) on the y-axis, or \((0,5)\) and \((0,-5)\). Plot these on the ellipse.

Key Concepts

Semi-Major Axis

The semi-major axis of an ellipse is one of its most crucial features. In an ellipse's equation, such as \[ \frac{x^{2}}{9} + \frac{y^{2}}{36} = 1, \]the semi-major axis can be identified depending on which denominator is larger. Here, since the larger denominator 36 is under the \(y^2\) term, the ellipse is vertically oriented, and the semi-major axis is determined by \(b = 6\). This means that the ellipse stretches farther along the y-axis.

• For ellipses, the axis aligned with the larger value in the equation is always the semi-major axis.
• If \(a^2\) were larger, the semi-major axis would be aligned with the x-axis.

The length of the semi-major axis is essentially the longest diameter of the ellipse, extending from the center to the outer edge on either side. In this case, starting from the center at the origin \((0, 0)\), it stretches 6 units in both the positive and negative y-directions. This distance provides the general shape and size of the ellipse.

Semi-Minor Axis

The semi-minor axis complements the semi-major axis by defining the ellipse's width. In the equation \[ \frac{x^{2}}{9} + \frac{y^{2}}{36} = 1, \]we find that \(a^2 = 9\), so \(a = 3\). This makes the semi-minor axis 3 units long and oriented along the x-axis. This axis is important for completing the shape of the ellipse, ensuring that it is not circular but elongated in one direction.

• A semi-minor axis should be thought of as the shorter radius of the ellipse.
• If the ellipse is horizontal, the semi-minor axis would still align with the secondary dimension, usually the y-axis.

In this ellipse, the minor axis indicates that from the center at the origin, it extends 3 units left and right along the x-axis. The orientation and length of the semi-minor axis help us to accurately graph and visualize the entire elliptic shape.

Foci of an Ellipse

The foci of an ellipse are pivotal points that contribute to the definition and properties of an ellipse. They lie along the major axis and help in understanding the ellipse's geometric properties. From the equation \[ \frac{x^{2}}{9} + \frac{y^{2}}{36} = 1, \]we compute the foci using the formula \( c = \sqrt{b^2 - a^2} \) to find \( c = \sqrt{36 - 9} = 5 \). Therefore, the foci are located at \((0, 5)\) and \((0, -5)\) along the y-axis because the ellipse is vertical.

• The foci are symmetrical with respect to the center of the ellipse.
• They lie on the major axis of the ellipse, determining its overall shape and direction.

Understanding the location of the foci is vital for applications involving ellipses in physics and engineering, as these points help maintain the balance and symmetry inherent in elliptical shapes. By plotting the foci, we reinforce the shape's characteristics and ensure the ellipse is correctly graphed.