Understanding the Ellipse: Graphing and Properties of the Equation (frac{(y-2)^2}{25} - frac{(x-3)^2}{5} 1)
In this article, we explore the equation of an ellipse and how to graph it by breaking down its components and identifying key features. We will delve into the concepts of center, foci, vertices, and the overall shape of the ellipse.
Introduction to the Equation
The given equation is:
(frac{(y-2)^2}{25} - frac{(x-3)^2}{5} 1)
This is the standard form of a hyperbola, not an ellipse. However, let's assume it was meant to represent an ellipse for the sake of this exploration. We can rewrite the equation in a standard form of an ellipse:
(frac{(y-2)^2}{25} frac{(x-3)^2}{5} 1)
Identifying the Center and Axes of the Ellipse
The general form of an ellipse centered at ((h, k)) is:
(frac{(y-k)^2}{a^2} frac{(x-h)^2}{b^2} 1)
From the given equation, we can identify:
h 3 (x-coordinate of the center) k 2 (y-coordinate of the center) 25 5 (length of the semi-major axis) 5 2.236 (length of the semi-minor axis)Center of the Ellipse
The center of the ellipse is at the point (3, 2).
Foci and Vertices of the Ellipse
In an ellipse, the foci and vertices are key points that lie along the major and minor axes.
Foci of the Ellipse
The foci of an ellipse lie along the major axis, and their distance from the center can be calculated using the formula:
(c sqrt{a^2 - b^2})
Substituting the values:
(c sqrt{25 - 5} sqrt{20} approx 4.472)
The foci are located at:
(3, 2 4.472) (3, 6.472) (upper focus) (3, 2 - 4.472) (3, -2.472) (lower focus)Vertices of the Ellipse
The vertices of the ellipse are the endpoints of the major and minor axes. They are as follows:
Major axis (along y-axis): (3, 2 5) and (3, 2 - 5) Minor axis (along x-axis): (3 2.236, 2) and (3 - 2.236, 2)Graphing the Ellipse
To graph the ellipse, follow these steps:
Plot the center at (3, 2). Plot the vertices on the major axis at (3, 7) and (3, -3). Plot the vertices on the minor axis at (5.236, 2) and (0.764, 2). Plot the foci at (3, 6.472) and (3, -2.472). Draw the ellipse through these points.Conclusion
The given equation was originally intended to be a hyperbola, but for the context of this exploration, it has been converted into an ellipse. The key features include the center at (3, 2), the major radius of 5, and the minor radius of approximately 2.236. Understanding these elements helps in accurately graphing and analyzing the ellipse.