2-%2B-(y%2B2)2-%3D-32.jpeg "(x-4)2 + (y+2)2 = 32")
Understanding the Equation: (x-4)² + (y+2)² = 32
The equation (x-4)² + (y+2)² = 32 represents a circle in the Cartesian coordinate system. Let's break down the components and understand what it tells us:
Key Concepts
- Standard Form: The equation is in the standard form of a circle: (x-h)² + (y-k)² = r², where:
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
- Center: In our equation, (h, k) = (4, -2). This tells us the center of the circle is at the point (4, -2).
- Radius: The square root of the constant term on the right side of the equation is the radius. Therefore, r = √32 = 4√2. The circle has a radius of 4√2 units.
Visualizing the Circle
Imagine plotting the point (4, -2) on a coordinate plane. Now draw a circle around this point with a radius of 4√2 units. This circle represents all the points (x, y) that satisfy the equation (x-4)² + (y+2)² = 32.
Finding Points on the Circle
You can find points on the circle by substituting different values for x or y into the equation and solving for the other variable. For example:
- If x = 4, then (4-4)² + (y+2)² = 32, which simplifies to (y+2)² = 32. Solving for y, we get y = -2 ± 4√2. This gives us two points on the circle: (4, -2 + 4√2) and (4, -2 - 4√2).
Applications
Understanding circles and their equations is crucial in various fields, including:
- Geometry: For analyzing geometric shapes and their properties.
- Physics: For describing the motion of objects in circular paths.
- Engineering: For designing circular structures and components.
- Computer Graphics: For generating and manipulating circular objects.
The equation (x-4)² + (y+2)² = 32 provides a compact and informative way to describe a specific circle, making it easier to analyze and work with in various contexts.