Problem 7 - Precalculus Mathematics for Calculus

Question

Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$\frac{x^{2}}{9}+\frac{(y+5)^{2}}{25}=1$$

Step-by-Step Solution

Answer

Center: (0, -5); Vertices: (0, 0) and (0, -10); Foci: (0, -1) and (0, -9); Major axis: 10; Minor axis: 6.

Step 1: Identify the Standard Form of the Ellipse

The given equation \( \frac{x^2}{9} + \frac{(y+5)^2}{25} = 1 \) is already in the standard form of an ellipse \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where the larger denominator corresponds to \( a^2 \). In this equation, \( a^2 = 25 \) and \( b^2 = 9 \).

Step 2: Calculate the Center

From the standard form, identify the center \((h, k)\) of the ellipse. Here, \(x\) is not shifted, meaning \( h = 0 \), and \(y\) has a shift of \(-5\), so \( k = -5 \). Therefore, the center is \( (0, -5) \).

Step 3: Determine the Vertices

The vertices lie along the major axis, which is vertical in this case because \( a^2 = 25 \) is greater than \( b^2 = 9 \). The vertices are at \((h, k \pm a)\). With \(a = \sqrt{25} = 5\), the vertices are \((0, -5 + 5) = (0, 0)\) and \((0, -5 - 5) = (0, -10)\).

Step 4: Find the Foci

For an ellipse, the foci lie along the major axis at a distance \(c\) from the center, where \(c\) is determined using \( c^2 = a^2 - b^2 \). Calculate \( c^2 = 25 - 9 = 16 \), so \( c = \sqrt{16} = 4 \). The foci are at \((0, -5 + 4) = (0, -1)\) and \((0, -5 - 4) = (0, -9)\).

Step 5: Determine the Lengths of Axes

The lengths of the major and minor axes are \( 2a \) and \( 2b \), respectively. With \(a = 5\) and \(b = 3\), the major axis length is \( 2 \times 5 = 10 \), and the minor axis length is \( 2 \times 3 = 6 \).

Step 6: Sketch the Ellipse

Plot the center \((0, -5)\) on a graph. Draw the major axis vertically, extending 5 units up and 5 units down from the center, reaching the vertices \((0, 0)\) and \((0, -10)\). Draw the minor axis horizontally, extending 3 units from the center, determining the ellipse's width. Mark the foci at \((0, -1)\) and \((0, -9)\). Sketch the ellipse accordingly.

Key Concepts

Standard Form of an Ellipse

To understand ellipses, it's essential to look at their standard equation form. The standard form of an ellipse is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). This equation helps you determine key features of the ellipse like its center, axes, and orientation.
- \(h\) and \(k\) are the coordinates of the ellipse's center. - The terms \(a^2\) and \(b^2\) represent the denominators associated with the ellipse's axis lengths. If the larger of these denominators is under the \(y\) term, the ellipse is vertically oriented; otherwise, it's horizontally oriented. For our equation \( \frac{x^2}{9} + \frac{(y+5)^2}{25} = 1 \), we see that 25 is larger than 9, so the major axis is vertical.

Elliptical Foci

The foci of an ellipse are significant because they help define the shape. They lie along the major axis and influence the curve of the ellipse, ensuring that the sum of the distances from the foci to any point on the ellipse remains constant. To find the foci, compute the distance \(c\) from the center using the formula \( c^2 = a^2 - b^2 \).
For the given equation, \(a^2 = 25\) and \(b^2 = 9\).

• Calculate \(c^2 = 25 - 9 = 16\).
• Thus, \(c = \sqrt{16} = 4\).

The foci will be at distances \(c\) above and below the center along the y-axis. This places the foci at coordinates \((0, -1)\) and \((0, -9)\). These positions tell us just how stretched the ellipse is along its major axis.

Major and Minor Axes

The major and minor axes of an ellipse correspond to its longest and shortest diameters, respectively. Understanding the lengths of these axes is crucial for sketching and grasping the ellipse's proportions. - The major axis length is determined by \(2a\). For our equation, \(a = \sqrt{25} = 5\), resulting in a major axis length of \(10\).- The minor axis length is \(2b\), with \(b = \sqrt{9} = 3\), resulting in a minor axis length of \(6\).These two axes are perpendicular, intersecting at the ellipse's center. From the center, the major axis extends equally in both directions along the y-axis in our example, while the minor axis stretches along the x-axis. Recognizing these dimensions helps visualize the complete shape and size of the ellipse.