Problem 26 - Thomas Calculus

Question

Exercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the \(x y\) -plane. In each case, find the ellipse's standard-form equation from the given information. $$ \text { Foci: }(0, \pm 4) \quad \text { Vertices: }(0, \pm 5) $$

Step-by-Step Solution

Answer

The standard-form equation of the ellipse is \(\frac{x^2}{9} + \frac{y^2}{25} = 1\).

Step 1: Understand the Given Information

We have an ellipse centered at the origin with foci located at \((0, \pm 4)\) and vertices located at \((0, \pm 5)\). The foci and vertices along the y-axis indicate a vertical ellipse.

Step 2: Determine Values of a and c

For vertical ellipses, the vertices are at \((0, \pm a)\). Hence, \(a = 5\) since the vertices are at \((0, \pm 5)\). The foci are given at \((0, \pm c)\), so \(c = 4\) since the foci are at \((0, \pm 4)\).

Step 3: Use the Relationship to Find b

In an ellipse, the relationship is given by \(c^2 = a^2 - b^2\). Substitute the known values: \(4^2 = 5^2 - b^2\).

Step 4: Calculate b

Solve the equation \(16 = 25 - b^2\). Rearrange to find \(b^2 = 25 - 16 = 9\), so \(b = 3\).

Step 5: Write the Standard Form Equation of the Ellipse

The standard form of a vertical ellipse centered at the origin is \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\). Substitute the values: \(\frac{x^2}{3^2} + \frac{y^2}{5^2} = 1\). Simplifying, the equation becomes \(\frac{x^2}{9} + \frac{y^2}{25} = 1\).

Key Concepts

Foci

The foci of an ellipse are like special points. They help define the shape of the ellipse. In our ellipse, the foci are located at

• (0, 4) and (0, -4)

This positioning tells us a couple of key things. The central point or the center is at the origin (0,0). Also, since these points are along the y-axis, it confirms that the ellipse is vertical. The distance between the center and each focus is 4 units. In ellipse terms, this distance is called "c". Remember, for an ellipse, the larger the distance between the foci, the more stretched out the ellipse will be. However, no matter how stretched, these foci always act like the 'balance points' of the shape.

Vertices

Vertices are the points where the ellipse is widest or tallest. In the given problem, the vertices are located at

• (0, 5) and (0, -5)

These points also lie along the y-axis, just like the foci. This further assures us that the ellipse is vertical. The distance from the center (0,0) to each vertex is 5 units. This length is denoted as "a" in ellipse mathematics. The distance to the vertices is always a bit more than the distance to the foci ('a' is always greater than 'c'). For ellipses, the vertices define how far the ellipse spreads from the center.

Equation of an Ellipse

The general formula for an ellipse is a starting point for writing its equation. If we know whether it's more stretched in the x-direction or y-direction, the equation gets simpler. For vertical ellipses like ours, the equation takes the form

• \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \)

where 'a' and 'b' represent the lengths from the center to the vertices and from the center to the co-vertices, respectively. In our exercise, we've figured out 'a' is 5 and 'b' is found to be 3 after some calculations. You'll find these lengths squared and placed under the respective axes in the equation. So,

• \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \)

This is the standard equation representing the ellipse.

Ellipse Centered at the Origin

When we say an ellipse is centered at the origin, it means the middle of the ellipse is at point (0,0) on the coordinate plane. It helps in writing the standard equation without extra terms.
Simply put, there are no shifts needed, either vertically or horizontally. It gives:

• The simplest form of the equation
• Ease of calculation and plotting

For any ellipse where the center is at the origin, calculations rely solely on the distances from this central point to the foci and the vertices. The equations remain straightforward as seen in our example:

• \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \)

It elegantly captures all the information about the ellipse's size and orientation relative to the origin.