Calculating The Major Axis Length Of An Ellipse

Understanding the properties of ellipses is a fundamental concept in mathematics, particularly in analytic geometry. Among these properties, the major axis holds significant importance. In this article, we will delve into the process of determining the length of the major axis of an ellipse, using the equation $\frac{(x-3)^2}{49}+\frac{(y-9)^2}{4}=1$ as a practical example. This comprehensive guide will not only provide a step-by-step solution but also enhance your understanding of ellipses and their key characteristics.

Understanding the Ellipse Equation

To determine the length of the major axis, we must first decipher the given equation: $\frac{(x-3)^2}{49}+\frac{(y-9)^2}{4}=1$. This equation represents an ellipse in its standard form. The standard form of an ellipse equation centered at $(h, k)$ is given by $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ for a horizontal ellipse (where $a > b$) and $\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$ for a vertical ellipse (where $a > b$). Here, $a$ represents the semi-major axis, $b$ represents the semi-minor axis, and the center of the ellipse is at the point $(h, k)$. By comparing the given equation with the standard form, we can extract crucial information about our specific ellipse. The values under the squared terms, 49 and 4, are particularly important as they directly relate to the lengths of the semi-major and semi-minor axes. It is crucial to correctly identify which value corresponds to $a^2$ and which corresponds to $b^2$ to accurately determine the major axis length. The larger value will always correspond to $a^2$, which is essential for finding the length of the major axis.

Identifying Key Parameters

In this section, we will focus on identifying the key parameters from the given equation: $\frac{(x-3)^2}{49}+\frac{(y-9)^2}{4}=1$. By carefully observing the equation, we can see that it closely resembles the standard form of an ellipse equation. The denominator under the $(x-3)^2$ term is 49, and the denominator under the $(y-9)^2$ term is 4. These values are crucial for determining the lengths of the semi-major and semi-minor axes. Recall that the standard form of an ellipse equation is $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ for a horizontal ellipse or $\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$ for a vertical ellipse. In our equation, $a^2$ and $b^2$ correspond to the denominators 49 and 4, respectively. Since 49 is greater than 4, we identify 49 as $a^2$ and 4 as $b^2$. This means that $a = \sqrt{49} = 7$ and $b = \sqrt{4} = 2$. The value of $a$ represents the length of the semi-major axis, and $b$ represents the length of the semi-minor axis. Furthermore, we can identify the center of the ellipse. The center $(h, k)$ corresponds to the values subtracted from $x$ and $y$ in the equation. In this case, $h = 3$ and $k = 9$, so the center of the ellipse is at the point $(3, 9)$. Identifying these key parameters – the semi-major axis length ($a$), the semi-minor axis length ($b$), and the center $(h, k)$ – is a crucial step in understanding the ellipse and its properties.

Calculating the Major Axis Length

Having identified the key parameters, the next step is to calculate the length of the major axis. The major axis is the longest diameter of the ellipse, and its length is directly related to the semi-major axis, denoted as $a$. Specifically, the length of the major axis is equal to $2a$. In the given equation, $\frac{(x-3)^2}{49}+\frac{(y-9)^2}{4}=1$, we determined that $a^2 = 49$, which means $a = \sqrt{49} = 7$. Therefore, the length of the semi-major axis is 7 units. To find the length of the major axis, we simply multiply the semi-major axis length by 2. Thus, the length of the major axis is $2 \times 7 = 14$ units. This calculation highlights the direct relationship between the semi-major axis and the major axis of an ellipse. Understanding this relationship is essential for quickly determining the major axis length once the value of $a$ is known. The major axis lies along the x-axis because the larger denominator (49) is under the $(x-3)^2$ term. This indicates that the ellipse is horizontally elongated, and the major axis spans 14 units along the horizontal direction.

Conclusion

In conclusion, we have successfully determined the length of the major axis for the ellipse represented by the equation $\frac{(x-3)^2}{49}+\frac{(y-9)^2}{4}=1$. By understanding the standard form of an ellipse equation, we identified the key parameters, including the semi-major axis length $a$. We found that $a = 7$, and subsequently calculated the major axis length as $2a = 14$ units. This process demonstrates the importance of recognizing the structure of the ellipse equation and its relationship to the ellipse's geometric properties. The major axis is a fundamental characteristic of an ellipse, and its length provides crucial information about the ellipse's shape and size. The ability to quickly and accurately determine the major axis length is a valuable skill in mathematics, particularly in analytic geometry and related fields. Understanding ellipses and their properties has numerous applications in various fields, including physics, engineering, and astronomy. From modeling planetary orbits to designing optical systems, the characteristics of ellipses play a crucial role. Mastering the concepts discussed in this guide will not only enhance your mathematical proficiency but also provide a solid foundation for exploring these real-world applications. The correct answer is A. 14 units.