2%2F25-%2B-(y-%E2%80%93-2)2%2F36-%3D-1.jpeg "(x + 3)2/25 + (y – 2)2/36 = 1")
Understanding the Equation: (x + 3)²/25 + (y – 2)²/36 = 1
This equation represents an ellipse, a geometric shape closely related to the circle. To understand it better, let's break down the components:
Key Features of the Equation
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Standard Form: The equation is written in the standard form of an ellipse centered at (h, k): (x - h)²/a² + (y - k)²/b² = 1
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Center: By comparing our equation to the standard form, we can identify the center of the ellipse: (-3, 2).
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Major and Minor Axes: The values under the x² and y² terms represent the squares of the semi-major axis (a) and semi-minor axis (b), respectively.
- a² = 25, so a = 5. The major axis is horizontal, with a length of 2a = 10.
- b² = 36, so b = 6. The minor axis is vertical, with a length of 2b = 12.
Visualizing the Ellipse
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Plot the Center: Locate the point (-3, 2) on the coordinate plane.
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Draw the Axes:
- From the center, move 5 units to the right and left to mark the endpoints of the major axis.
- From the center, move 6 units up and down to mark the endpoints of the minor axis.
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Sketch the Ellipse: Draw a smooth curve connecting the endpoints of the axes, forming the shape of an ellipse.
Important Properties
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Foci: The ellipse has two foci, points inside the ellipse that play a crucial role in its definition. The distance from any point on the ellipse to the two foci is constant. The foci lie on the major axis, and their distance from the center is given by: c = √(a² - b²). In this case, c = √(25 - 36) = √(-11) which is not a real number. This indicates that the ellipse is not centered at the origin, making the calculation of the foci more complex.
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Eccentricity: Eccentricity (e) measures how much the ellipse deviates from a circle. It is calculated as e = c/a. In this case, e is imaginary due to the imaginary value of c.
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Symmetry: The ellipse is symmetrical about both the major and minor axes.
Applications
Ellipses find applications in various fields:
- Astronomy: Planetary orbits are often elliptical.
- Engineering: Ellipses are used in designing arches, bridges, and other structures.
- Optics: The shape of a mirror used in a telescope can be elliptical to focus light.
- Mathematics: Ellipses are a fundamental concept in geometry and calculus.
By understanding the equation and its components, we can easily visualize and analyze the properties of this ellipse.