Finding The Center Of An Ellipse A Step-by-Step Guide

In the captivating realm of mathematics, ellipses stand as elegant geometric figures, captivating mathematicians and enthusiasts alike. An ellipse, a close cousin to the circle, is defined as the set of all points where the sum of the distances from two fixed points, known as the foci, is constant. These graceful curves find applications in various fields, from planetary orbits to engineering designs.

The equation presented, $\frac{(y+5)^2}{121}+\frac{(x-9)^2}{49}=1$, unveils an ellipse poised on the coordinate plane. To fully grasp the essence of this ellipse, we must first delve into the general equation that governs these captivating curves. The standard form equation of an ellipse centered at point $(h, k)$ is expressed as:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

where:

• $(h, k)$ represents the coordinates of the ellipse's center.
• $a$ denotes the semi-major axis, the distance from the center to the farthest point on the ellipse.
• $b$ signifies the semi-minor axis, the distance from the center to the closest point on the ellipse.

This equation serves as a compass, guiding us to decipher the ellipse's critical attributes, including its center, orientation, and dimensions. The values of $a$ and $b$ dictate the ellipse's shape, with a larger difference between them resulting in a more elongated ellipse. The position of the major axis, determined by whether $a^2$ is under the $(x-h)^2$ or $(y-k)^2$ term, reveals the ellipse's orientation, either horizontal or vertical, respectively.

Our primary goal is to pinpoint the center of the ellipse represented by the equation $\frac{(y+5)^2}{121}+\frac{(x-9)^2}{49}=1$. This equation, though seemingly complex, holds the key to unlocking the ellipse's center. To accomplish this, we embark on a comparative analysis, juxtaposing our given equation with the standard form equation of an ellipse.

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

$\frac{(y+5)^2}{121}+\frac{(x-9)^2}{49}=1$

By carefully aligning the terms, we can discern the values of $h$ and $k$, which directly correspond to the coordinates of the ellipse's center. Observe that the term $(x - h)^2$ in the standard form aligns with $(x - 9)^2$ in our equation. This congruence unveils the value of $h$ as 9. Similarly, the term $(y - k)^2$ in the standard form aligns with $(y + 5)^2$ in our equation. Rewriting $(y + 5)$ as $(y - (-5))$ unveils the value of $k$ as -5.

Therefore, the center of the ellipse is located at the point $(h, k) = (9, -5)$. This revelation marks a significant step in our understanding of the ellipse, providing a crucial reference point for further analysis.

Having pinpointed the center of the ellipse, our quest for knowledge extends to deciphering other crucial parameters. The values of $a$ and $b$, representing the semi-major and semi-minor axes, respectively, hold valuable insights into the ellipse's dimensions and orientation.

In our equation, $\frac{(y+5)^2}{121}+\frac{(x-9)^2}{49}=1$, we observe that the denominator under the $(y + 5)^2$ term is 121, which corresponds to $b^2$. Taking the square root of 121, we obtain $b = 11$. This signifies that the semi-major axis, the longer axis of the ellipse, has a length of 11 units.

Similarly, the denominator under the $(x - 9)^2$ term is 49, which corresponds to $a^2$. Taking the square root of 49, we obtain $a = 7$. This signifies that the semi-minor axis, the shorter axis of the ellipse, has a length of 7 units.

The larger value of $b$ compared to $a$ indicates that the ellipse is vertically oriented, with its major axis aligned along the vertical direction. This understanding of the ellipse's orientation is crucial for visualizing its shape and position in the coordinate plane.

With the center, semi-major axis, and semi-minor axis in our grasp, we can now visualize the ellipse in its entirety. Imagine a graceful oval, centered at the point (9, -5), stretching 11 units vertically and 7 units horizontally. This mental image brings the equation to life, transforming abstract symbols into a tangible geometric form.

The center, (9, -5), serves as the ellipse's anchor, the point around which its symmetry revolves. The semi-major axis, with its length of 11 units, dictates the ellipse's vertical extent, while the semi-minor axis, with its length of 7 units, governs its horizontal span. The vertical orientation, stemming from the larger value of $b$, imparts an elongated appearance to the ellipse, stretching it along the y-axis.

This visualization is not merely an aesthetic exercise; it deepens our understanding of the ellipse's properties and behavior. It allows us to anticipate its interactions with other geometric figures and to predict its role in various applications.

The center of the ellipse holds a position of paramount importance, serving as a reference point for all other attributes. It acts as the midpoint of both the major and minor axes, dictating the ellipse's symmetry and balance. Understanding the center's coordinates is fundamental to deciphering the ellipse's position and orientation within the coordinate plane.

The center also plays a crucial role in determining the foci of the ellipse, the two fixed points that define its shape. The distance between the center and each focus is directly related to the lengths of the semi-major and semi-minor axes. By knowing the center and the axes' lengths, we can precisely locate the foci, further enhancing our understanding of the ellipse's geometry.

In practical applications, the center often serves as a reference point for design and analysis. For instance, in architecture, the center of an elliptical arch might be used as a reference point for structural calculations. In astronomy, the center of a planet's elliptical orbit is a key parameter for understanding its motion around the sun.

In this comprehensive exploration, we have successfully unveiled the center of the ellipse represented by the equation $\frac{(y+5)^2}{121}+\frac{(x-9)^2}{49}=1$. Through a meticulous comparison with the standard form equation of an ellipse, we have identified the center's coordinates as (9, -5). This revelation serves as a cornerstone for our understanding of the ellipse, paving the way for further analysis and application.

Moreover, we have delved into the significance of the center, highlighting its role as a reference point for the ellipse's symmetry, orientation, and foci. We have also emphasized the center's practical applications in various fields, underscoring its importance in both theoretical and real-world contexts.

By mastering the art of identifying the center of an ellipse, we equip ourselves with a valuable tool for navigating the fascinating world of conic sections and their diverse applications. This understanding empowers us to appreciate the elegance and versatility of ellipses in mathematics and beyond.

Therefore, the correct answer is D. (9, -5).