%5E2y-4)%5E2%3D169.jpeg "(x-9)^2y-4)^2=169")
Unveiling the Secrets of (x-9)^2(y-4)^2 = 169
The equation (x-9)^2(y-4)^2 = 169 might look intimidating at first glance, but it actually represents a fascinating geometric shape with a rich history. Let's explore its secrets.
A Hidden Ellipse
This equation describes an ellipse, a closed curve that resembles a stretched circle. Here's why:
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Standard Ellipse Form: The standard form of an ellipse centered at (h,k) is: (x-h)^2/a^2 + (y-k)^2/b^2 = 1
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Transforming our Equation: To see the ellipse, we need to rearrange our equation slightly: (x-9)^2 / 13^2 + (y-4)^2 / 13^2 = 1
Now it aligns perfectly with the standard form!
Key Features
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Center: The center of the ellipse is at the point (9, 4).
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Major and Minor Axes: Since both denominators are 13^2, the ellipse has equal major and minor axes, which means it's a circle with a radius of 13 units.
Visualizing the Ellipse
Imagine a circle with a radius of 13 units centered at the point (9, 4). This circle is the solution to the equation (x-9)^2(y-4)^2 = 169.
Applications
Elliptical shapes have numerous real-world applications:
- Planetary Orbits: Planets orbit the sun in elliptical paths.
- Optics: Elliptical reflectors are used in telescopes and other optical devices.
- Architecture: Elliptical arches are a common architectural feature.
Conclusion
The equation (x-9)^2(y-4)^2 = 169 might seem complex, but it ultimately represents a simple and elegant geometric shape – a circle with a radius of 13 units. By understanding the equation's relationship to the standard form of an ellipse, we gain insight into the fascinating world of conic sections and their wide-ranging applications.