Finding The Equation Of An Ellipse With Minor Axis 10 And Foci At (3,6) And (7,6)

In the realm of analytic geometry, ellipses hold a special place, characterized by their unique shape and mathematical properties. An ellipse can be defined as the set of all points such that the sum of the distances from two fixed points (the foci) is constant. This fundamental property dictates the ellipse's equation and its overall structure. Understanding the relationship between an ellipse's parameters, such as its major and minor axes, foci, and center, is crucial for determining its equation. This article delves into the process of finding the equation of an ellipse given specific characteristics, such as the length of the minor axis and the location of the foci. We will walk through the necessary steps, emphasizing the underlying concepts and formulas. By the end of this exploration, you will be well-equipped to tackle similar problems involving ellipses and their equations.

Understanding Ellipses: A Comprehensive Guide

Before diving into the specific problem, it's essential to establish a strong foundation in the properties of ellipses. An ellipse is essentially a stretched circle, and its shape is determined by two key parameters: the major axis and the minor axis. The major axis is the longest diameter of the ellipse, passing through the center and both foci, while the minor axis is the shortest diameter, perpendicular to the major axis and also passing through the center. The foci, as mentioned earlier, are two fixed points inside the ellipse that play a crucial role in its definition. The distance between the foci is a significant factor in determining the ellipse's shape; the closer the foci, the more circular the ellipse becomes.

The center of the ellipse is the midpoint of both the major and minor axes, serving as the ellipse's central point of symmetry. The standard form equation of an ellipse centered at $(h, k)$ depends on whether the major axis is horizontal or vertical. For a horizontal major axis, the equation is given by $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, where 'a' is the semi-major axis (half the length of the major axis) and 'b' is the semi-minor axis (half the length of the minor axis). Conversely, for a vertical major axis, the equation is $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$, with 'a' and 'b' still representing the semi-major and semi-minor axes, respectively. The relationship between 'a', 'b', and the distance from the center to each focus, denoted by 'c', is given by the equation $c^2 = a^2 - b^2$. This equation is a cornerstone in solving ellipse-related problems.

Understanding these fundamental concepts and formulas is paramount to accurately determining the equation of an ellipse when provided with its characteristics. We will now apply these principles to the specific problem at hand, systematically unraveling the solution.

Problem Breakdown: Minor Axis Length and Foci Location

Let's revisit the problem statement: we need to identify the equation of an ellipse with a minor axis of length 10 and foci located at $(3,6)$ and $(7,6)$. The first crucial step is to extract the key information provided. The minor axis length of 10 immediately tells us that the semi-minor axis, 'b', is half of this length, which is 5. The foci coordinates, $(3,6)$ and $(7,6)$, provide valuable insights into the ellipse's orientation and center. Notice that the y-coordinates of both foci are the same, indicating that the major axis is horizontal. This is because the foci always lie on the major axis.

To find the center of the ellipse, we can calculate the midpoint of the foci. The midpoint formula, given by $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$, is a fundamental tool in coordinate geometry. Applying this formula to the foci $(3,6)$ and $(7,6)$, we get the center as $(\frac{3+7}{2}, \frac{6+6}{2}) = (5,6)$. Thus, we now know the center of the ellipse, $(h, k) = (5,6)$. The distance between the foci is another critical piece of information. The distance formula, $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$, allows us to calculate the distance between two points. In this case, the distance between the foci $(3,6)$ and $(7,6)$ is $\sqrt{(7-3)^2 + (6-6)^2} = \sqrt{4^2} = 4$. This distance is equal to $2c$, where 'c' is the distance from the center to each focus. Therefore, $2c = 4$, and $c = 2$.

We have now successfully determined the semi-minor axis, 'b', the center of the ellipse, $(h, k)$, and the distance from the center to each focus, 'c'. The next step involves utilizing the relationship between 'a', 'b', and 'c' to find the semi-major axis, 'a'.

Calculating the Semi-Major Axis and Forming the Equation

Recall the equation that connects the semi-major axis (a), the semi-minor axis (b), and the distance from the center to each focus (c): $c^2 = a^2 - b^2$. We have already established that $b = 5$ and $c = 2$. Substituting these values into the equation, we get $2^2 = a^2 - 5^2$, which simplifies to $4 = a^2 - 25$. Adding 25 to both sides, we have $a^2 = 29$. Therefore, $a = \sqrt{29}$. This value represents the semi-major axis of the ellipse.

Now that we have determined the semi-major axis 'a', the semi-minor axis 'b', and the center $(h, k)$, we can construct the equation of the ellipse. Since we established earlier that the major axis is horizontal, we will use the standard form equation for an ellipse with a horizontal major axis: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$. Substituting the values we found, we have $h = 5$, $k = 6$, $a^2 = 29$, and $b^2 = 5^2 = 25$. Plugging these values into the equation, we get $\frac{(x-5)^2}{29} + \frac{(y-6)^2}{25} = 1$.

This equation represents the ellipse with a minor axis of length 10 and foci located at $(3,6)$ and $(7,6)$. By systematically breaking down the problem, identifying key parameters, and applying the relevant formulas, we have successfully derived the equation of the ellipse. It's important to note the significance of understanding the underlying geometric principles and the relationships between the ellipse's parameters in solving such problems. Let's compare our result with the given options to confirm the solution.

Solution Verification and Conclusion

Comparing the derived equation, $\frac{(x-5)^2}{29} + \frac{(y-6)^2}{25} = 1$, with the options provided in the problem, we can see that option A, $\frac{(x-5)^2}{25} + \frac{(y-6)^2}{29} = 1$, and option B, $\frac{(x+5)^2}{29} + \frac{(y+6)^2}{25} = 1$, are not the correct equations. Therefore, there seems to be a discrepancy in the provided options. The correct equation, based on our calculations, is $\frac{(x-5)^2}{29} + \frac{(y-6)^2}{25} = 1$.

This exercise highlights the importance of careful calculation and a thorough understanding of the concepts involved. Even if the provided options do not match the derived solution, the process of systematically analyzing the problem and applying the relevant formulas allows us to arrive at the correct answer. The key takeaways from this problem include the understanding of the relationship between the ellipse's parameters, the use of the midpoint and distance formulas, and the application of the standard form equation of an ellipse.

In conclusion, by carefully analyzing the given information, applying the relevant formulas, and systematically deriving the equation, we can confidently determine the equation of an ellipse given its minor axis length and foci locations. This problem serves as a valuable illustration of the power of analytical geometry in solving geometric problems.