Ellipse Equation: (x – 3)²/25 + (y + 1)²/16 = 1
In this article, we will discuss the ellipse equation (x – 3)²/25 + (y + 1)²/16 = 1
. We will explore the graph of the equation, its center, vertices, and foci.
Graph of the Equation
The equation (x – 3)²/25 + (y + 1)²/16 = 1
represents an ellipse centered at (3, -1)
. The graph of the equation is as follows:
<h2 align="center"> <img src="https://latex.artofproblemsolving.com/8/3/1/831c5fa5c4b37fba91a71cfe61d2c1e5a4b42f96.png" alt="Graph of the equation" /> </h2>
Center of the Ellipse
The center of the ellipse is (3, -1)
. This is evident from the equation, where the center is shifted 3 units to the right and 1 unit down from the origin.
Vertices of the Ellipse
The vertices of the ellipse are the points where the ellipse intersects the axes. In this case, the vertices are (8, -1)
and (-2, -1)
.
Foci of the Ellipse
The foci of the ellipse are the points (5.48, -1)
and (0.52, -1)
. These points are calculated using the formula c = sqrt(a² - b²)
, where a
is the semi-major axis and b
is the semi-minor axis.
Properties of the Ellipse
- The semi-major axis is 5 units (
a = 5
). - The semi-minor axis is 4 units (
b = 4
). - The eccentricity of the ellipse is
e = 0.6
.
In conclusion, the equation (x – 3)²/25 + (y + 1)²/16 = 1
represents an ellipse centered at (3, -1)
with vertices (8, -1)
and (-2, -1)
, and foci (5.48, -1)
and (0.52, -1)
.