Problem 3 - Algebra and Trigonometry

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

The center is at (0, -5), vertices at (0, 0) and (0, -10), foci at (0, -1) and (0, -9), and axes lengths are 10 (major) and 6 (minor).

Step 1: Identify the Standard Form of the Ellipse Equation

The given ellipse equation \( \frac{x^2}{9} + \frac{(y+5)^2}{25} = 1 \) is already in the standard form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). In this case, \( h = 0 \), \( k = -5 \), \( a^2 = 9 \), and \( b^2 = 25 \).

Step 2: Determine the Center of the Ellipse

Using the values of \( h \) and \( k \) from the equation, the center of the ellipse \((h, k)\) is at the point \((0, -5)\).

Step 3: Find the Lengths of the Major and Minor Axes

The major axis length is \(2b\) because \(b^2\) (25) is larger than \(a^2\) (9), meaning the major axis is vertical. Therefore, \(b = 5\). The length of the major axis is \(2 \times 5 = 10\). The minor axis length is \(2a\), with \(a = 3\), so \(2 \times 3 = 6\).

Step 4: Locate the Vertices of the Ellipse

Since the major axis is vertical, the vertices are offset from the center vertically. They are at \((0, -5 + 5) = (0, 0)\) and \((0, -5 - 5) = (0, -10)\).

Step 5: Calculate the Foci of the Ellipse

The distance from the center to each focus \(c\) is calculated using the equation \( c^2 = b^2 - a^2 \). Therefore, \( c^2 = 25 - 9 = 16 \), giving \( c = 4 \). The foci, being on the vertical axis, are located at \((0, -5 + 4) = (0, -1)\) and \((0, -5 - 4) = (0, -9)\).

Step 6: Sketch the Ellipse

To sketch the ellipse, plot the center at \((0, -5)\), the vertices at \((0, 0)\) and \((0, -10)\), and the foci at \((0, -1)\) and \((0, -9)\). Draw the ellipse with a vertical major axis, using the calculated lengths of the axes as guides.

Key Concepts

Conic Sections

Conic sections are shapes you get when a plane cuts through a cone. There are four main types: circles, ellipses, parabolas, and hyperbolas. Each type has unique properties and equations. Ellipses, like in our exercise, are particularly interesting because they look like stretched circles. You can find them in orbits of planets around the sun! In math, you recognize them from their specific equation formats. This specific ellipse lies in the category of conic sections due to its special equation form which describes it in the coordinate system.

Major Axis

The major axis is the longest diameter of the ellipse. It runs through the center and both foci. For our ellipse, the major axis is vertical because the larger value of the denominators in its equation is under the \(y\) part. Hence, the major axis length is determined by \(b\) in cases where \(b^2 > a^2\).

• In the given equation, \(b = 5\) since \(b^2 = 25\).
• Thus, the length of the major axis is \(2b = 10\).

The endpoints of the major axis are the vertices of the ellipse, showing how far the ellipse stretches along this axis.

Minor Axis

The minor axis is the shorter diameter of the ellipse, perpendicular to the major axis. For our particular ellipse, it lies horizontally.

• The length is given by \(2a\).
• Here, \(a = 3\) because \(a^2 = 9\).
• Consequently, the minor axis length is \(2 \times 3 = 6\).

This minor axis helps give the ellipse its characteristic stretched shape, indicating how wide the ellipse is.

Foci of Ellipse

An ellipse has two special points called foci (plural of focus). Each focus is on the major axis at a point \(c\) from the ellipse's center. They are essential for understanding ellipses as they help define the shape.To locate them:

• Use the formula \(c^2 = b^2 - a^2\).
• For our example, \(c^2 = 25 - 9 = 16\), leading to \(c = 4\).
• The foci of this ellipse are then located at \( (0, -5 + 4) = (0, -1) \) and \( (0, -5 - 4) = (0, -9) \).

The closeness or distance of foci from the center gives the ellipse its shape, with a more pronounced stretch for greater distances.

Vertices of Ellipse

Vertices are the points where the ellipse intersects the major axis. They represent the ellipse's widest points along this axis.For the given ellipse:

• The vertices are located at \( (h, k \pm b) \).
• As calculated, they are \( (0, 0) \) and \( (0, -10) \), symmetrically placed around the center.

These vertices visually define the elongation of the ellipse and help in sketching the elliptical shape accurately.