Problem 42 - Thomas Calculus

Question

You may wish to review Section 1.2 before solving Exercises \(39-56\) The ellipse \(\left(x^{2} / 9\right)+\left(y^{2} / 25\right)=1\) is shifted 3 units to the left and 2 units down to generate the ellipse $$ \frac{(x+3)^{2}}{9}+\frac{(y+2)^{2}}{25}=1 $$ $$ \begin{array}{l}{\text { a. Find the foci, vertices, and center of the new ellipse. }} \\ {\text { b. Plot the new foci, vertices, and center, and sketch in the new }} \\ {\text { ellipse. }}\end{array} $$

Step-by-Step Solution

Answer

Center: (-3,-2); Vertices: (-3,3), (-3,-7); Foci: (-3,2), (-3,-6).

Step 1: Identify the New Center of the Ellipse

The transformation \( (x+3)^2/9 + (y+2)^2/25 = 1 \) indicates that the original center \( (0,0) \) has been shifted 3 units to the left and 2 units down. Thus, the new center is \( (-3, -2) \).

Step 2: Confirm the Equation is in Standard Form

The equation \( \frac{(x+3)^2}{9} + \frac{(y+2)^2}{25} = 1 \) confirms that it represents an ellipse in standard form: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). Here, \( a = 3 \) and \( b = 5 \). This tells us the ellipse is oriented with the major axis along the y-direction.

Step 3: Calculate the Vertices

The vertices along the major y-axis can be calculated by moving \( b = 5 \) units up and down from the center. Since the center is \( (-3, -2) \), the vertices are at \( (-3, -2+5) = (-3, 3) \) and \( (-3, -2-5) = (-3, -7) \).

Step 4: Calculate the Minor Axis Vertices

For the minor axis, move \( a = 3 \) units right and left from the center. The vertices of the minor axis are \( (-3+3, -2) = (0, -2) \) and \( (-3-3, -2) = (-6, -2) \).

Step 5: Calculate the Foci

The distance to the foci \( c \) on the major axis can be determined using the formula \( c^2 = b^2 - a^2 \). Thus, \( c^2 = 5^2 - 3^2 = 25 - 9 = 16 \), so \( c = 4 \). The foci are \( (-3, -2+4) = (-3, 2) \) and \( (-3, -2-4) = (-3, -6) \).

Step 6: Sketch the Ellipse and Plot Key Points

To sketch the ellipse:1. Plot the center at \( (-3, -2) \).2. Plot the vertices at \( (-3, 3) \) and \( (-3, -7) \) for the major axis, and \( (0, -2) \) and \( (-6, -2) \) for the minor axis.3. Plot the foci at \( (-3, 2) \) and \( (-3, -6) \).4. Draw the ellipse through these points following its shape.

Key Concepts

Ellipse Standard Form

An ellipse is a fascinating shape in mathematics, and to write its equation in a standard form gives us a lot of information about its size, orientation, and position.
The standard form of an ellipse's equation is: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]Here,

• \((h, k)\) is the center of the ellipse.
• \(a\) is the distance from the center to a vertex along the horizontal axis, also called the semi-major axis if \(a > b\).
• \(b\) is the distance from the center to a vertex along the vertical axis, also known as the semi-minor axis if \(b > a\).

In our exercise, after translation, the ellipse equation becomes \[ \frac{(x+3)^2}{9} + \frac{(y+2)^2}{25} = 1 \]Indicating that the center has moved to \((-3, -2)\). Here, \(a = 3\), indicating the ellipse elongates 3 units horizontally and \(b = 5\), which means it stretches 5 units vertically. It helps to determine the ellipse stretches more along the vertical due to \(b > a\).

Foci and Vertices of Ellipse

To explore more about the shape and orientation of an ellipse, we calculate its vertices and foci.

**Vertices**
A vertex is a point where the ellipse is widest or highest, based on which axis is longer.

• For the given translated ellipse, \(b\) is greater than \(a\), meaning the vertical direction is longer, hence it's the major axis.
• The vertices along the major axis are located by moving \(b = 5\) units up and down from the center point \((-3, -2)\). This computation results in vertices at \((-3, 3)\) and \((-3, -7)\).
• Conversely, vertices along the horizontal (minor) axis are computed by moving \(a = 3\) units right and left, forming points at \((0, -2)\) and \((-6, -2)\).

**Foci**
Foci are key points located along the major axis inside the ellipse. These points help in defining the shape.

• Calculated via: \(c^2 = b^2 - a^2\). For our ellipse, \(c^2 = 25 - 9 = 16\) gives \(c = 4\).
• The translated foci are positioned \(4\) units above and below the center, hence they reside at \((-3, 2)\) and \((-3, -6)\).

The vertices and foci assist in accurately sketching the dimensions of the ellipse on a coordinate plane.

Coordinate Geometry Transformations

Understanding how the coordinate geometry transformations affect the position of shapes is essential.
For an ellipse, translating or moving it involves changing its location without altering its shape.

**Translation**

• The exercise comprises translating the ellipse \(3\) units left, which deducts \(3\) from the x-coordinates, indicating by \((x+3)\).
• Additionally, the ellipse shifts \(2\) units down, an adjustment visible in the y-coordinate as \((y+2)\).

This translation repositions the ellipse's center from \((0, 0)\) to \((-3, -2)\). The rest of the significant points (foci and vertices) are subsequently recalculated.

**Effect on Ellipse Properties**

• This type of transformation retains the shape, size, and orientation of the ellipse.
• Only its position in the plane changes, allowing you to re-assess its center, vertices, and foci based on its new location.

Understanding these transformations helps to solve geometry problems effectively and visualize solutions more clearly.