Finding Major Axis Ends And Foci Of The Ellipse $49x^2 + 16y^2 = 784$

Introduction

In the realm of mathematics, ellipses hold a significant place as one of the fundamental conic sections. Understanding their properties, such as the major axis and foci, is crucial in various fields, including physics, astronomy, and engineering. This article delves into the process of finding the ends of the major axis and foci of a given ellipse equation: $49x^2 + 16y^2 = 784$. We will explore the standard form of an ellipse equation, identify the key parameters, and apply the relevant formulas to determine the required points. Mastering these concepts not only enhances your mathematical proficiency but also provides a solid foundation for tackling more complex problems involving ellipses and other conic sections. So, let's embark on this mathematical journey and unravel the mysteries of the ellipse.

Understanding the Ellipse Equation

To begin, we must recognize the given equation, $49x^2 + 16y^2 = 784$, represents an ellipse. The general form of an ellipse centered at the origin is given by:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

where a and b are the semi-major and semi-minor axes, respectively. If a > b, the major axis is horizontal, and if b > a, the major axis is vertical. The foci are points inside the ellipse that are equidistant from the center, and their distance from the center is denoted by c. The relationship between a, b, and c is given by:

$$c^2 = |a^2 - b^2|$$

In our case, the equation $49x^2 + 16y^2 = 784$ needs to be transformed into the standard form to identify a and b. Dividing both sides of the equation by 784, we get:

$$\frac{49x^2}{784} + \frac{16y^2}{784} = 1$$

Simplifying the fractions, we have:

$$\frac{x^2}{16} + \frac{y^2}{49} = 1$$

Now, the equation is in the standard form, and we can easily identify that $a^2 = 16$ and $b^2 = 49$. Therefore, a = 4 and b = 7. Since b > a, the major axis is vertical. This means the ellipse is stretched along the y-axis, and the vertices (ends of the major axis) will lie on the y-axis. The understanding of this fundamental concept is crucial for finding the ends of the major axis and the foci accurately.

Finding the Ends of the Major Axis

Since the major axis is vertical, the ends of the major axis, also known as the vertices, will be located at $(0, ext{\pm}b)$. We have already determined that b = 7. Therefore, the ends of the major axis are:

$$(0, 7) ext{ and } (0, -7)$$

The major axis is the longest diameter of the ellipse, and its length is 2b, which in this case is 2 * 7 = 14 units. The vertices are the points where the ellipse intersects its major axis. Understanding the orientation of the major axis, whether horizontal or vertical, is key to correctly identifying the vertices. A clear grasp of the ellipse's geometry is essential for visualizing the location of these points.

Calculating the Foci

To find the foci, we need to calculate the distance c from the center to each focus. We use the formula:

$$c^2 = |a^2 - b^2|$$

Substituting the values of $a^2$ and $b^2$, we get:

$$c^2 = |16 - 49|$$

$$c^2 = |-33|$$

$$c^2 = 33$$

Therefore, c = $\sqrt{33}$.

Since the major axis is vertical, the foci will be located on the y-axis at $(0, ext{\pm}c)$. Thus, the foci are:

$$(0, \sqrt{33}) ext{ and } (0, -\sqrt{33})$$

The foci are critical points within the ellipse that define its shape. The sum of the distances from any point on the ellipse to the two foci is constant. This property is often used in the definition of an ellipse and has practical applications in fields like optics and acoustics. The ability to calculate the foci accurately is a vital skill in understanding the ellipse's properties and its relationship to other conic sections.

Summarizing the Results

In summary, for the ellipse given by the equation $49x^2 + 16y^2 = 784$, we have found:

• Ends of the Major Axis: $(0, 7)$ and $(0, -7)$
• Foci: $(0, \sqrt{33})$ and $(0, -\sqrt{33})$

These points are crucial characteristics of the ellipse, defining its shape and orientation in the coordinate plane. The process involved converting the given equation to the standard form, identifying the semi-major and semi-minor axes, and applying the appropriate formulas to calculate the vertices and foci. This methodical approach is essential for solving similar problems and gaining a deeper understanding of ellipses.

Conclusion

Finding the ends of the major axis and foci of an ellipse is a fundamental exercise in conic sections. By converting the equation to standard form, identifying the key parameters a and b, and applying the formula for c, we can accurately determine these points. This process highlights the importance of understanding the relationship between the equation of an ellipse and its geometric properties. The ability to analyze and solve such problems is a valuable skill in mathematics and its applications. Understanding the ellipse and its properties, such as the major axis and foci, provides a strong foundation for further exploration of conic sections and their applications in various fields. The mathematical rigor applied in this process not only enhances problem-solving skills but also fosters a deeper appreciation for the elegance and interconnectedness of mathematical concepts.

Further Exploration

To further enhance your understanding of ellipses, consider exploring the following topics:

• Eccentricity of an Ellipse: Learn how eccentricity affects the shape of an ellipse, ranging from a circle (eccentricity = 0) to a highly elongated shape (eccentricity close to 1).
• Directrices of an Ellipse: Investigate the concept of directrices and their relationship to the foci and points on the ellipse.
• Applications of Ellipses: Discover real-world applications of ellipses in fields such as optics, astronomy (planetary orbits), and engineering (bridge design).
• Hyperbolas and Parabolas: Extend your knowledge of conic sections by studying hyperbolas and parabolas and comparing their properties to those of ellipses.

By delving deeper into these areas, you will gain a more comprehensive understanding of conic sections and their significance in mathematics and beyond. This continuous learning will empower you to tackle more complex problems and appreciate the beauty of mathematical concepts.