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.