Problem 23 - Contemporary Precalculus

Question

In Exercises \(15-24,\) use the techniques of Examples 4 and 5 to graph the equation in a suitable square viewing window. $$9 x^{2}+5 y^{2}=45$$

Step-by-Step Solution

Answer

Answer: The center of the ellipse is at the origin (0,0). The vertices are located at (0, ±3), and the co-vertices are located at (±√5, 0).

Step 1: Divide by 45

First, we will divide the entire equation by 45 to make it easier to find the ellipse's significant points. $$\frac{9 x^{2}}{45}+\frac{5 y^{2}}{45}=1 \Rightarrow \frac{x^{2}}{5}+\frac{y^{2}}{9}=1$$

Step 2: Find the center

The center of the ellipse can be found using the formula \((h, k)\), where h and k are the x and y coordinates. In this case, \(h=0\) and \(k=0\), so the center of the ellipse is at the origin \((0,0)\).

Step 3: Determine the length of the major and minor axes

The length of the major axis (2a) and the minor axis (2b) can be found by comparing the coefficients in the equation to the standard ellipse formula \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\). In this case, \(a^{2} = 5\) and \(b^{2} = 9\), so the lengths of the major and minor axes are 2b (2sqrt(9)) = 6 and 2a (2sqrt(5)) = 4.47, respectively.

Step 4: Find the vertices and co-vertices

The vertices are located along the major (longer) axis. In this case, since the center is at \((0,0)\), the vertices are going to be \((0, \pm b) = (0, \pm 3)\). The co-vertices are located along the minor (shorter) axis, and their coordinates are \((\pm a, 0) = (\pm\sqrt{5},0)\).

Step 5: Graph the equation in a suitable viewing window

Now that we have the center, vertices, and co-vertices, we can begin graphing the equation in a suitable square viewing window. - Place the center at \((0, 0)\). - Plot the vertices at \((0, \pm 3)\). - Plot the co-vertices at \((\pm\sqrt{5}, 0)\). - Draw the ellipse by sketching a smooth curve connecting the vertices and co-vertices. The suitable square viewing window should have axes limits of at least \(x = [-1\sqrt{5}, 1\sqrt{5}]\) and \(y = [-3,3]\) to accommodate the entire ellipse.

Key Concepts

Equation of Ellipse

An ellipse is a shape that resembles a stretched circle. Its equation usually looks like this: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). In this equation, \(a\) and \(b\) are the lengths of the major and minor axes, respectively. Each value is squared, and the sum should equal 1. This format is known as the standard form of an ellipse centered at the origin

• \(x^2/a^2\) represents the horizontal component.
• \(y^2/b^2\) represents the vertical component.

For our equation, after a little reshuffling from \(9x^2 + 5y^2 = 45\), we got \(\frac{x^2}{5} + \frac{y^2}{9} = 1\) by dividing through by 45. This brings it to a standard form, making it easier to identify key components for graphing.
Additionally, by comparing this with the standard form, we identify that in this exercise, \(a^2\) is 5 and \(b^2\) is 9.

Major and Minor Axes

Within an ellipse, two fundamental lines are crucial: the major and minor axes. The major axis is the longest line that spans the ellipse, while the minor axis is the shortest.
To find the lengths of these axes in the equation \(\frac{x^2}{5} + \frac{y^2}{9} = 1\):

• The square root of the larger denominator determines the length of the semi-major axis \(b = \sqrt{9} = 3\), hence the full major axis is \(2b = 6\).
• The square root of the smaller denominator gives us the semi-minor axis \(a = \sqrt{5} \approx 2.24\), resulting in the minor axis length being \(2a \approx 4.47\).

The longer axis in our problem is vertical since \(b^2 = 9\) (larger than \(a^2 = 5\)). This means the ellipse is stretched more along the y-axis.

Vertices and Co-vertices

Vertices and co-vertices are specific points on the ellipse that align with its axes. They help in accurately plotting the shape.
For the equation \(\frac{x^2}{5} + \frac{y^2}{9} = 1\), the vertices and co-vertices are:

• **Vertices**: Located on the major axis. With the center at the origin, the vertices are positioned at \((0, \pm 3)\) because the major axis is vertical.
• **Co-vertices**: Found on the minor axis. Here, they're located at \((\pm \sqrt{5}, 0)\), or approximately \((\pm 2.24, 0)\).

These points are crucial for sketching the ellipse.
Start at the center and plot these key points to form a boundary, ensuring each axis points are marked accurately. Then, connect them smoothly in a loop to bring the graph of the ellipse to life.