Ellipse Equation With Vertices (2,9) And (2,-5) And Foci (2,5) And (2,-1) Detailed Solution
Introduction: Delving into the World of Ellipses
In the fascinating realm of analytic geometry, the ellipse stands out as a captivating conic section, characterized by its unique shape and properties. An ellipse can be defined as the set of all points where the sum of the distances to two fixed points, known as the foci, is constant. Understanding the key parameters of an ellipse, such as its vertices, foci, center, major axis, and minor axis, is crucial for deciphering its equation and graphical representation. This article aims to guide you through the process of determining the equation of an ellipse given its vertices and foci, shedding light on the underlying principles and calculations involved.
In this particular case, we are presented with an ellipse whose vertices are located at the points (2, 9) and (2, -5), and whose foci are situated at (2, 5) and (2, -1). Our mission is to unravel the equation that precisely describes this ellipse. By carefully analyzing the given information and applying the standard equation of an ellipse, we will embark on a step-by-step journey to unveil the solution.
Before we dive into the specifics of this problem, let's take a moment to review the fundamental concepts of ellipses. An ellipse possesses two axes of symmetry: the major axis and the minor axis. The major axis is the longer axis, passing through the foci and the vertices, while the minor axis is the shorter axis, perpendicular to the major axis and passing through the center of the ellipse. The vertices are the endpoints of the major axis, and the center is the midpoint of the major axis. The foci, as mentioned earlier, are two fixed points within the ellipse that play a crucial role in defining its shape.
The distance between the center and each vertex is denoted by 'a', which represents the semi-major axis. Similarly, the distance between the center and each endpoint of the minor axis is denoted by 'b', representing the semi-minor axis. The distance between the center and each focus is denoted by 'c'. These three parameters, a, b, and c, are interconnected by the fundamental relationship: c^2 = a^2 - b^2. This equation forms the cornerstone of our calculations and will be instrumental in determining the equation of the ellipse.
Now, with a solid grasp of the foundational concepts, let's embark on the quest to solve the problem at hand. We will meticulously analyze the given information, calculate the key parameters, and ultimately arrive at the equation that represents the ellipse with vertices at (2, 9) and (2, -5) and foci at (2, 5) and (2, -1).
Step 1: Pinpointing the Center of the Ellipse
The center of an ellipse serves as its pivotal point, the very heart around which the curve gracefully unfolds. In essence, the center is the midpoint of the line segment connecting the vertices, offering us a crucial starting point in our quest to define the ellipse's equation. To pinpoint the center, we employ the midpoint formula, a simple yet powerful tool that allows us to locate the exact middle ground between two points.
Given the vertices of our ellipse, (2, 9) and (2, -5), we can readily apply the midpoint formula. The midpoint formula, a cornerstone of coordinate geometry, states that the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by ((x1 + x2)/2, (y1 + y2)/2). This formula, derived from the principles of averaging coordinates, provides us with a direct and precise method for locating the center of our ellipse.
Plugging in the coordinates of our vertices, we have (x1, y1) = (2, 9) and (x2, y2) = (2, -5). Substituting these values into the midpoint formula, we get:
Center = ((2 + 2)/2, (9 + (-5))/2) Center = (4/2, 4/2) Center = (2, 2)
Thus, we have successfully located the center of our ellipse at the point (2, 2). This pivotal point will serve as the reference for all subsequent calculations, allowing us to determine the ellipse's dimensions and orientation.
Step 2: Deciphering the Value of 'a' (Semi-major Axis)
Understanding the ellipse requires knowing about semi-major axis, denoted by 'a', is a fundamental parameter that dictates the ellipse's overall size and shape. It represents the distance from the center of the ellipse to one of its vertices, effectively defining half the length of the major axis. In essence, the semi-major axis provides us with a measure of the ellipse's extension along its longest dimension.
To determine the value of 'a' for our ellipse, we leverage the coordinates of the center, which we previously calculated to be (2, 2), and the coordinates of one of the vertices, say (2, 9). The distance between these two points represents the semi-major axis. We can calculate this distance using the distance formula, a versatile tool that allows us to measure the separation between any two points in a coordinate plane.
The distance formula, rooted in the Pythagorean theorem, states that the distance between two points (x1, y1) and (x2, y2) is given by √((x2 - x1)^2 + (y2 - y1)^2). This formula, a cornerstone of Euclidean geometry, provides us with a precise method for calculating the distance between the center and a vertex, thereby revealing the value of 'a'.
Plugging in the coordinates of the center (2, 2) and the vertex (2, 9) into the distance formula, we get:
a = √((2 - 2)^2 + (9 - 2)^2) a = √(0^2 + 7^2) a = √49 a = 7
Therefore, we have successfully determined that the semi-major axis, 'a', of our ellipse is 7 units. This value signifies the extent of the ellipse along its major axis and will be crucial in formulating the ellipse's equation.
Step 3: Unveiling the Value of 'c' (Distance from Center to Focus)
In the intricate dance of an ellipse's properties, understanding foci is key, 'c', which represents the distance from the center of the ellipse to one of its foci. This parameter provides us with insights into the ellipse's eccentricity, a measure of how elongated or circular the ellipse is. A larger value of 'c' implies a more elongated ellipse, while a smaller value suggests a more circular shape.
To determine the value of 'c' for our ellipse, we utilize the coordinates of the center, which we previously established as (2, 2), and the coordinates of one of the foci, say (2, 5). The distance between these two points represents the distance 'c'. As in the previous step, we employ the distance formula, a reliable tool for measuring the separation between points in a coordinate plane.
Applying the distance formula to the center (2, 2) and the focus (2, 5), we get:
c = √((2 - 2)^2 + (5 - 2)^2) c = √(0^2 + 3^2) c = √9 c = 3
Thus, we have successfully unveiled the value of 'c' for our ellipse, which is 3 units. This parameter, representing the distance from the center to a focus, provides us with valuable information about the ellipse's shape and will be essential in determining the remaining parameter, 'b'.
Step 4: Calculating the Value of 'b' (Semi-minor Axis)
The semi-minor axis, denoted by 'b', is another crucial parameter that defines the ellipse's shape, complementing the role of the semi-major axis. It represents the distance from the center of the ellipse to an endpoint of the minor axis, effectively defining half the length of the minor axis. While the semi-major axis governs the ellipse's extension along its longest dimension, the semi-minor axis governs its extension along its shortest dimension.
To calculate the value of 'b' for our ellipse, we turn to the fundamental relationship that connects the parameters 'a', 'b', and 'c': c^2 = a^2 - b^2. This equation, a cornerstone of ellipse geometry, allows us to determine 'b' if we know the values of 'a' and 'c'. We have already calculated 'a' to be 7 and 'c' to be 3, so we can now substitute these values into the equation and solve for 'b'.
Substituting the values of 'a' and 'c' into the equation, we get:
3^2 = 7^2 - b^2 9 = 49 - b^2
Now, we rearrange the equation to isolate b^2:
b^2 = 49 - 9 b^2 = 40
Finally, we take the square root of both sides to find the value of 'b':
b = √40
Therefore, we have successfully calculated the value of the semi-minor axis, 'b', which is √40 units. This parameter, along with the semi-major axis 'a', fully defines the dimensions of our ellipse.
Step 5: Constructing the Equation of the Ellipse
With all the necessary parameters in hand, we are now poised to construct the equation of our ellipse. The standard form equation is key to understanding elliptical shapes. The standard form of an ellipse equation depends on whether the major axis is horizontal or vertical. Since the vertices and foci of our ellipse share the same x-coordinate (2), we know that the major axis is vertical. Therefore, we will use the standard form equation for an ellipse with a vertical major axis:
(x - h)^2 / b^2 + (y - k)^2 / a^2 = 1
where (h, k) represents the center of the ellipse, 'a' is the semi-major axis, and 'b' is the semi-minor axis. We have already determined that the center of our ellipse is (2, 2), the semi-major axis 'a' is 7, and the semi-minor axis 'b' is √40. Now, we simply substitute these values into the standard form equation.
Substituting the values, we get:
(x - 2)^2 / (√40)^2 + (y - 2)^2 / 7^2 = 1
Simplifying the equation, we have:
(x - 2)^2 / 40 + (y - 2)^2 / 49 = 1
Thus, we have successfully constructed the equation of the ellipse with vertices at (2, 9) and (2, -5) and foci at (2, 5) and (2, -1). The equation is (x - 2)^2 / 40 + (y - 2)^2 / 49 = 1.
Conclusion: The Ellipse Revealed
In this article, we embarked on a journey to unravel the equation of an ellipse given its vertices and foci. By systematically applying the principles of analytic geometry, we navigated through the steps of identifying the center, determining the semi-major and semi-minor axes, and ultimately constructing the equation that precisely describes the ellipse. The equation (x - 2)^2 / 40 + (y - 2)^2 / 49 = 1 stands as the culmination of our efforts, a testament to the power of mathematical reasoning and problem-solving.
Understanding the equation of an ellipse not only allows us to represent it mathematically but also provides us with a deeper appreciation for its geometric properties. The ellipse, with its graceful curves and unique characteristics, finds applications in various fields, from astronomy to architecture. By mastering the techniques presented in this article, you can confidently tackle similar problems and further expand your understanding of conic sections and analytic geometry.
Final Answer:
The correct answer is B.
