Ellipse Foci Calculation A Step-by-Step Guide

In the realm of analytic geometry, the equation $\frac{(y-8)^2}{49} + \frac{(x-1)^2}{36} = 1$ unveils the presence of an ellipse, a captivating geometric figure. This equation, a symphony of numbers and variables, encapsulates the very essence of the ellipse's shape, size, and orientation within the Cartesian plane. To truly grasp the nature of this ellipse, we must embark on a journey of deciphering its components, unraveling the secrets hidden within its mathematical form. This exploration will lead us to the heart of the ellipse, where we will pinpoint the approximate locations of its foci, those enigmatic points that govern the ellipse's curvature and symmetry.

Delving into the Ellipse Equation

The equation $\frac{(y-8)^2}{49} + \frac{(x-1)^2}{36} = 1$ adheres to the standard form of an ellipse equation, which serves as a blueprint for understanding its key characteristics. This standard form, a cornerstone of ellipse analysis, takes the guise of $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ for a horizontal ellipse and $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ for a vertical ellipse. Here, (h, k) marks the ellipse's center, the very epicenter around which the ellipse gracefully curves. The parameters 'a' and 'b' dictate the ellipse's dimensions, with 'a' representing the semi-major axis, the longer radius, and 'b' representing the semi-minor axis, the shorter radius. These axes, like the warp and weft of a fabric, define the ellipse's overall shape and extent.

In our specific equation, $\frac{(y-8)^2}{49} + \frac{(x-1)^2}{36} = 1$, we can discern the center of the ellipse as (1, 8), a pivotal point in the Cartesian plane. Furthermore, we observe that the denominator under the (y - 8)^2 term, 49, is larger than the denominator under the (x - 1)^2 term, 36. This disparity signals that the major axis of the ellipse aligns vertically, akin to a skyscraper reaching for the sky. The semi-major axis, 'a', is the square root of 49, which is 7, while the semi-minor axis, 'b', is the square root of 36, which is 6. These values, like architectural blueprints, define the ellipse's dimensions, with the major axis stretching 7 units vertically from the center and the minor axis extending 6 units horizontally.

Unveiling the Foci

The foci, those elusive points nestled within the ellipse, hold the key to understanding its curvature and symmetry. These points, like celestial anchors, exert their influence on the ellipse's shape, dictating the graceful curves that define its form. To locate the foci, we embark on a mathematical quest, employing the relationship between the semi-major axis ('a'), the semi-minor axis ('b'), and the distance from the center to each focus ('c'). This relationship, a fundamental principle of ellipse geometry, is expressed by the equation $c^2 = a^2 - b^2$.

In our case, we have already established that a = 7 and b = 6. Plugging these values into the equation, we get $c^2 = 7^2 - 6^2 = 49 - 36 = 13$. Taking the square root of both sides, we find that c = √13, which is approximately 3.6. This value, 3.6, represents the distance from the center of the ellipse to each focus. Since the major axis is vertical, the foci will lie along a vertical line passing through the center of the ellipse. Therefore, to find the coordinates of the foci, we add and subtract this distance, 3.6, from the y-coordinate of the center, which is 8.

Thus, the foci are located at approximately (1, 8 + 3.6) and (1, 8 - 3.6), which translate to (1, 11.6) and (1, 4.4). These points, like celestial beacons, mark the foci of the ellipse, guiding its curvature and symmetry. However, among the given options, only option C, (1, 4.4), aligns with our calculated foci locations. This discrepancy invites us to scrutinize the options, to discern which one provides the most accurate approximation.

Identifying the Approximate Locations

Having determined the theoretical locations of the foci as approximately (1, 11.6) and (1, 4.4), we now turn our attention to the provided options, seeking the closest match. Option A, (-8, -4.6) and (-8, 2.6), immediately stands out as an unlikely candidate. Its coordinates bear no resemblance to our calculated values, with x-coordinates of -8 contrasting sharply with our derived x-coordinate of 1. Moreover, the y-coordinates deviate significantly from our calculated values of 11.6 and 4.4. Thus, option A can be confidently discarded.

Option B, (-2.6, -8) and (4.6, 8), presents a slightly more intriguing scenario. While the y-coordinates include 8, matching the y-coordinate of the ellipse's center, the x-coordinates, -2.6 and 4.6, diverge considerably from our calculated x-coordinate of 1. This disparity casts doubt on the accuracy of option B, making it an unlikely contender for the true foci locations.

Option C, (1, 4.4) and (1, 11.6), emerges as the most promising candidate. Its x-coordinate, 1, perfectly aligns with our calculated x-coordinate for the foci. Furthermore, one of its y-coordinates, 4.4, precisely matches our derived y-coordinate for one of the foci. While the other y-coordinate, 11.6, might seem slightly off, it's important to remember that we're dealing with approximations. Rounding 11.6 to the nearest tenth, we get 11.6, which aligns perfectly with our calculated value.

Therefore, based on our analysis, option C, (1, 4.4) and (1, 11.6), provides the most accurate approximation of the foci's locations. This option, like a guiding star, leads us to the true heart of the ellipse, revealing the points that govern its curvature and symmetry.

In conclusion, by meticulously analyzing the equation $\frac{(y-8)^2}{49} + \frac{(x-1)^2}{36} = 1$, we have successfully navigated the realm of ellipses, pinpointing the approximate locations of its foci. Through a combination of mathematical principles and careful calculations, we have demonstrated that the foci are best represented by the points (1, 4.4) and (1, 11.6). This journey, like a voyage of discovery, has deepened our understanding of ellipses, those captivating geometric figures that grace the world of mathematics.

Keywords: ellipse, foci, equation, center, major axis, minor axis, semi-major axis, semi-minor axis, approximation, coordinates, Cartesian plane, analytic geometry.

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Find the approximate locations of the foci of the ellipse represented by the equation $\frac{(y-8)^2}{49} + \frac{(x-1)^2}{36} = 1$. Round the coordinates to the nearest tenth.

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Finding Foci of an Ellipse Step-by-Step Guide