%5E2%2By%5E2%3D9.jpeg "(x-1)^2+y^2=9")
Exploring the Equation: (x-1)^2 + y^2 = 9
The equation (x-1)^2 + y^2 = 9 represents a circle in the Cartesian coordinate system. Let's break down why and explore its key properties.
Understanding the Standard Form
The standard form of a circle's equation is:
(x - h)^2 + (y - k)^2 = r^2
Where:
- (h, k) is the center of the circle
- r is the radius of the circle
Analyzing Our Equation
Comparing our equation, (x-1)^2 + y^2 = 9, to the standard form, we can identify:
- Center: (h, k) = (1, 0)
- Radius: r^2 = 9, so r = 3
Key Features of the Circle
- Center: The circle is centered at the point (1, 0).
- Radius: The circle has a radius of 3 units.
- Location: The circle is located in the first quadrant of the coordinate plane, as its center is in the first quadrant and its radius extends outward.
Visualizing the Circle
To visualize the circle, you can:
- Plot the center: Mark the point (1, 0) on a graph.
- Draw the radius: From the center, measure 3 units in all directions (up, down, left, right) and mark these points.
- Connect the points: Connect the marked points to form a smooth circle.
Applications
Circles are fundamental geometric shapes with many applications in:
- Geometry: Calculating area, circumference, and other geometric properties.
- Trigonometry: Defining angles and trigonometric functions.
- Physics: Modeling circular motion and wave propagation.
- Engineering: Designing circular structures and components.
By understanding the equation and its properties, we can effectively analyze and work with this circle and its applications.