%5E2%2B(y%2B3)%5E2%3D4.jpeg "(x-1)^2+(y+3)^2=4")
Understanding the Equation (x-1)² + (y+3)² = 4
The equation (x-1)² + (y+3)² = 4 represents a circle in the Cartesian coordinate system. Let's break down the key elements and how to interpret this equation:
Standard Form of a Circle Equation
The general form of a circle's equation is:
(x - h)² + (y - k)² = r²
Where:
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
Applying to Our Equation
Comparing our given equation (x-1)² + (y+3)² = 4 to the standard form, we can identify the following:
- Center: (h, k) = (1, -3)
- Radius: r² = 4, so r = 2
Graphical Representation
This information allows us to easily graph the circle:
- Locate the center: Plot the point (1, -3) on the coordinate plane.
- Draw the radius: From the center, mark points 2 units away in all directions (up, down, left, right).
- Connect the points: Draw a smooth curve connecting the points, forming a circle.
Key Features
Here are some key features of the circle represented by (x-1)² + (y+3)² = 4:
- Center: The circle is centered at the point (1, -3).
- Radius: The circle has a radius of 2 units.
- Circumference: The circumference of the circle is 2πr = 4π units.
- Area: The area of the circle is πr² = 4π square units.
Applications
Understanding circle equations like (x-1)² + (y+3)² = 4 is essential in various fields, including:
- Geometry: For analyzing geometric shapes and their properties.
- Physics: For describing circular motion and orbits.
- Engineering: For designing circular structures and components.
- Computer Graphics: For creating and manipulating circular objects in digital environments.
By understanding the equation's components and its graphical representation, we can gain valuable insights into the properties and applications of circles.