%5E2%2B(y-6)%5E2%3D121.jpeg "(x+5)^2+(y-6)^2=121")
Unveiling the Circle: (x+5)² + (y-6)² = 121
The equation (x+5)² + (y-6)² = 121 represents a circle, a fundamental shape in geometry. Let's explore its properties and understand its meaning.
The Standard Form of a Circle
The general equation for a circle is:
(x - h)² + (y - k)² = r²
Where:
- (h, k) represents the coordinates of the center of the circle.
- r represents the radius of the circle.
Analyzing the Equation
Comparing our equation (x+5)² + (y-6)² = 121 to the standard form, we can identify the following:
- Center: The center of the circle is at (-5, 6), since (h, k) = (-5, 6).
- Radius: The radius of the circle is 11 units, since r² = 121, and r = √121 = 11.
Visualizing the Circle
To visualize the circle, we can plot its center at (-5, 6) and draw a circle with a radius of 11 units extending outward from the center in all directions.
Key Properties
- Symmetry: The circle is symmetrical about both the x-axis and y-axis.
- Circumference: The circumference of the circle is 2πr = 2π(11) = 22π units.
- Area: The area of the circle is πr² = π(11)² = 121π square units.
Applications
Understanding the equation of a circle has applications in various fields, including:
- Geometry: Analyzing circles and their relationships with other geometric shapes.
- Physics: Describing circular motion and trajectories.
- Computer Graphics: Representing circular objects in computer graphics and animations.
In conclusion, the equation (x+5)² + (y-6)² = 121 defines a circle with a specific center and radius, offering valuable information about its properties and allowing for its visual representation and analysis.