2%2F25-%2B-(y-%2B-1)2%2F16-%3D-1.jpeg "(x – 3)2/25 + (y + 1)2/16 = 1")
Understanding the Equation: (x – 3)2/25 + (y + 1)2/16 = 1
This equation represents an ellipse, a fundamental geometric shape. Let's break down the equation and explore its key features.
Standard Form of an Ellipse
The general standard form for an ellipse centered at (h, k) is:
(x - h)2/a2 + (y - k)2/b2 = 1
Where:
- (h, k) represents the center of the ellipse.
- a represents the distance from the center to the vertices along the major axis.
- b represents the distance from the center to the vertices along the minor axis.
Analyzing our Equation
Comparing our given equation (x – 3)2/25 + (y + 1)2/16 = 1 to the standard form, we can identify the following:
- Center: (h, k) = (3, -1)
- a2 = 25, so a = 5 (This is the distance from the center to the vertices along the horizontal axis)
- b2 = 16, so b = 4 (This is the distance from the center to the vertices along the vertical axis)
Key Features of the Ellipse
Based on the information we have gathered:
- Center: The ellipse is centered at the point (3, -1).
- Major Axis: The major axis is horizontal, with length 2a = 10.
- Minor Axis: The minor axis is vertical, with length 2b = 8.
- Vertices: The vertices are located 5 units to the left and right of the center: (3 - 5, -1) = (-2, -1) and (3 + 5, -1) = (8, -1).
- Co-vertices: The co-vertices are located 4 units above and below the center: (3, -1 + 4) = (3, 3) and (3, -1 - 4) = (3, -5).
Visualizing the Ellipse
To visualize the ellipse, you can plot the center, vertices, and co-vertices on a coordinate plane. Then, sketch a smooth curve that passes through these points, forming the shape of an ellipse.
Remember: Ellipses are symmetrical about both their major and minor axes. This symmetry aids in creating a more accurate visual representation.
Conclusion
By understanding the standard form and analyzing the given equation, we can easily identify the center, major and minor axes, and vertices of the ellipse represented by (x – 3)2/25 + (y + 1)2/16 = 1. This knowledge allows us to accurately visualize and understand the properties of this specific ellipse.