Finding The Equation Of An Ellipse Focus (0, 3) And Minor Axis Length 8

Finding the equation of an ellipse given specific parameters is a common problem in analytic geometry. In this article, we will tackle the problem of determining the equation of an ellipse whose focus is located at the point (0, 3) and the length of the minor axis is 8. This involves understanding the standard form of an ellipse equation, the relationship between its parameters (foci, semi-major axis, semi-minor axis), and applying the given information to derive the equation. Let's dive into the detailed steps.

Understanding the Ellipse Equation and Properties

Before we start, let’s establish the foundational knowledge required to solve this problem. An ellipse is a conic section defined as the locus of points such that the sum of the distances from two fixed points (foci) is constant. The standard equation of an ellipse depends on whether the major axis is horizontal or vertical. The general forms are:

• Horizontal Major Axis: (x-h)²/a² + (y-k)²/b² = 1
• Vertical Major Axis: (x-h)²/b² + (y-k)²/a² = 1

Where:

• (h, k) is the center of the ellipse.
• a is the semi-major axis (half the length of the major axis).
• b is the semi-minor axis (half the length of the minor axis).
• The foci are located at a distance c from the center, where c² = a² - b² for a horizontal ellipse and c² = a² - b² for a vertical ellipse.

Key Properties and Relationships

1. Center: The center of the ellipse is the midpoint of the line segment connecting the foci. If we know the foci, we can find the center.
2. Major and Minor Axes: The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. Their lengths are 2a and 2b, respectively.
3. Foci: The foci are two points inside the ellipse that define its shape. For an ellipse centered at the origin, the foci are at (±c, 0) for a horizontal ellipse and (0, ±c) for a vertical ellipse.
4. Relationship Between a, b, and c: The relationship between the semi-major axis (a), the semi-minor axis (b), and the distance from the center to a focus (c) is given by c² = a² - b².

Understanding these properties and relationships is crucial for solving problems involving ellipses. In our case, we are given the focus and the length of the minor axis, which we will use to find the equation of the ellipse.

Step-by-Step Solution

Given the focus at (0, 3) and the length of the minor axis is 8, let's find the equation of the ellipse. Here's how we can approach this problem step-by-step:

1. Determine the Center of the Ellipse

Since we have only one focus given at (0, 3), we need to infer the location of the other focus and the center. The major axis passes through the foci, and the center is the midpoint of the segment connecting the foci. Given the symmetry of the ellipse, the center must lie on the y-axis, and the other focus will be equidistant from the center as the given focus. Assuming the ellipse is centered at (0, k), the given focus (0, 3) implies that the other focus must be at (0, 2k - 3).

If we assume that the ellipse is centered at the origin (0, 0), the foci would be at (0, ±c). However, since one focus is at (0, 3), the ellipse cannot be centered at the origin unless there is another focus at (0, -3). This implies that c = 3. If the center is not at the origin, the foci would be at (h, k ± c). Given the focus (0, 3), we can infer that the x-coordinate of the center h = 0, and the center lies on the y-axis. To simplify the problem, we'll assume the center is at (0, 0), implying the other focus is at (0, -3).

2. Determine the Orientation of the Ellipse

The foci are located along the y-axis (0, 3) and (0, -3), which means the major axis is vertical. Therefore, the standard form of the ellipse equation will be:

(x²/b²) + (y²/a²) = 1

where a is the semi-major axis and b is the semi-minor axis.

3. Find the Semi-Minor Axis (b)

The length of the minor axis is given as 8. The semi-minor axis b is half of this length:

b = 8 / 2 = 4

So, b = 4, and b² = 16.

4. Find the Distance from the Center to the Focus (c)

The distance c from the center (0, 0) to the focus (0, 3) is 3:

c = 3

Thus, c² = 9.

5. Find the Semi-Major Axis (a)

We use the relationship c² = a² - b² to find a. Substituting the values we have:

9 = a² - 16
a² = 9 + 16
a² = 25
a = 5

So, a = 5, and a² = 25.

6. Write the Equation of the Ellipse

Now we have all the necessary values to write the equation of the ellipse. Since the major axis is vertical, the equation is:

(x²/b²) + (y²/a²) = 1

Substitute a² = 25 and b² = 16:

(x²/16) + (y²/25) = 1

7. Verification and Conclusion

The equation of the ellipse is (x²/16) + (y²/25) = 1. This represents an ellipse centered at the origin with a vertical major axis. The foci are at (0, ±3), and the length of the minor axis is 2b = 8, which matches the given information.

Alternative Scenario: Center Not at the Origin

Suppose the center of the ellipse is not at the origin. Let’s consider the center at (0, k). One focus is at (0, 3). Let the other focus be at (0, f). The center (0, k) is the midpoint of the foci, so:

k = (3 + f) / 2

We also know that the distance from the center to each focus is c. If we let the center be at (0, k), then:

c = |3 - k| = |k - f|

From the given information, the length of the minor axis is 8, so b = 4 and b² = 16. Let's assume the semi-major axis is a. We have the relationship c² = a² - b². Since the major axis is along the y-axis, the equation of the ellipse would be:

(x²/16) + ((y - k)²/a²) = 1

If we assume the center is at (0, 4), then c = |3 - 4| = 1. Then:

1 = a² - 16
a² = 17

The equation of the ellipse would be:

(x²/16) + ((y - 4)²/17) = 1

This alternative scenario illustrates that the center’s location significantly impacts the final equation. The center's position must be determined accurately to derive the correct equation.

Common Mistakes and How to Avoid Them

When finding the equation of an ellipse, several common mistakes can lead to incorrect solutions. Here’s a rundown of frequent errors and strategies to avoid them:

1. Misidentifying the Major and Minor Axes

• Mistake: Confusing the orientation of the major and minor axes, leading to an incorrect standard form equation.
• How to Avoid: Determine the major axis orientation by examining the foci. If the foci lie along the x-axis, the major axis is horizontal; if they lie along the y-axis, the major axis is vertical. Always visualize the ellipse with the given information.

2. Incorrectly Applying the Relationship Between a, b, and c

• Mistake: Using the wrong formula (e.g., a² = b² + c² instead of c² = a² - b²) or miscalculating values.
• How to Avoid: Remember that c² = a² - b² applies to both horizontal and vertical ellipses. Double-check your calculations and ensure you correctly substitute the values.

3. Assuming the Center is Always at the Origin

• Mistake: Automatically assuming the center of the ellipse is at (0, 0) without proper justification.
• How to Avoid: Use the given information about the foci or other parameters to determine the center. If only one focus is given, infer the location of the other focus and then find the midpoint, which gives the center.

4. Algebraic Errors

• Mistake: Making mistakes in algebraic manipulations, such as squaring terms or solving for variables.
• How to Avoid: Write down each step clearly and double-check your work. Use a systematic approach and avoid skipping steps to minimize errors.

5. Not Verifying the Solution

• Mistake: Failing to verify the final equation against the given conditions (focus, minor axis length, etc.).
• How to Avoid: After finding the equation, check that the foci calculated from your equation match the given foci and that the lengths of the axes are consistent with the given information.

6. Confusion with Hyperbolas

• Mistake: Mixing up the formulas and properties of ellipses with those of hyperbolas, which have similar but distinct equations.
• How to Avoid: Clearly understand the differences in the standard forms and relationships between a, b, and c for ellipses and hyperbolas. For ellipses, c² = a² - b², whereas for hyperbolas, c² = a² + b².

7. Not Considering Alternative Solutions

• Mistake: Overlooking other possible scenarios or solutions, such as different center locations that could satisfy the conditions.
• How to Avoid: Be open to alternative solutions and carefully analyze the problem statement to ensure you haven't missed any possibilities.

By being aware of these common mistakes and implementing the strategies to avoid them, you can improve your accuracy and confidence in solving ellipse-related problems.

Conclusion

In conclusion, finding the equation of an ellipse given specific parameters such as the focus and the length of the minor axis involves a systematic approach. Understanding the properties of ellipses, the relationship between the semi-major axis, semi-minor axis, and the distance to the foci, and carefully applying these concepts are crucial. We determined that the equation of the ellipse with a focus at (0, 3) and a minor axis length of 8, centered at the origin, is (x²/16) + (y²/25) = 1. Additionally, considering alternative scenarios, such as the center not being at the origin, provides a more comprehensive understanding of ellipse equations. By avoiding common mistakes and verifying the solution, one can confidently solve similar problems in analytic geometry. This detailed exploration ensures a solid grasp of the process and enhances problem-solving skills in mathematics.