2%2F25-%E2%80%93-(y-%E2%80%93-2)2%2F144-%3D-1.jpeg "(x + 4)2/25 – (y – 2)2/144 = 1")
Equation of an Ellipse: (x + 4)2/25 – (y – 2)2/144 = 1
Introduction
In mathematics, an ellipse is a type of conic section that is defined as the set of points in a plane that satisfy a specific quadratic equation. In this article, we will explore the equation (x + 4)2/25 – (y – 2)2/144 = 1, which represents an ellipse in the Cartesian coordinate system.
The Equation of an Ellipse
The equation (x + 4)2/25 – (y – 2)2/144 = 1 is a quadratic equation in two variables, x and y. This equation can be written in the standard form of an ellipse as:
(x - h)2/a2 - (y - k)2/b2 = 1
where (h, k) is the center of the ellipse, and a and b are the lengths of the semi-axes.
Center of the Ellipse
To find the center of the ellipse, we can rewrite the equation as:
((x + 4) / 5)2 - ((y - 2) / 12)2 = 1
This shows that the center of the ellipse is at (-4, 2).
Semi-Axes of the Ellipse
The lengths of the semi-axes of the ellipse can be found by examining the coefficients of the x and y terms. In this case, a = 5 and b = 12.
Graph of the Ellipse
The graph of the ellipse (x + 4)2/25 – (y – 2)2/144 = 1 is an ellipse centered at (-4, 2) with semi-axes of length 5 and 12.
Properties of the Ellipse
The ellipse (x + 4)2/25 – (y – 2)2/144 = 1 has several properties, including:
- Center: The center of the ellipse is at
(-4, 2). - Semi-axes: The lengths of the semi-axes are 5 and 12.
- Orientation: The major axis is vertical, meaning that the longer axis is parallel to the y-axis.
- Vertices: The vertices of the ellipse are at
(-4, -10)and(-4, 14).
Conclusion
In conclusion, the equation (x + 4)2/25 – (y – 2)2/144 = 1 represents an ellipse with center (-4, 2), semi-axes of length 5 and 12, and a vertical orientation. Understanding the properties of this ellipse is essential in various mathematical and real-world applications.
