%5E2%2B(y-7)%5E2%3D100.jpeg "(x+3)^2+(y-7)^2=100")
Exploring the Equation (x+3)^2 + (y-7)^2 = 100
The equation (x+3)^2 + (y-7)^2 = 100 represents a circle in the Cartesian coordinate system. Let's break down the components and understand what it signifies.
The Standard Form of a Circle
The general equation of a circle in standard form is:
(x - h)^2 + (y - k)^2 = r^2
Where:
- (h, k) represents the coordinates of the center of the circle
- r represents the radius of the circle
Analyzing the Equation
Comparing our given equation (x+3)^2 + (y-7)^2 = 100 to the standard form, we can identify the following:
- Center: The center of the circle is at (-3, 7). Notice how the signs of the constants within the parentheses are reversed when determining the center coordinates.
- Radius: The radius of the circle is 10. This is because 100 is the square of the radius (r^2 = 100).
Visualizing the Circle
To visualize the circle, we can plot the center point (-3, 7) and then draw a circle with a radius of 10 units around this point.
Applications
The equation of a circle has numerous applications in mathematics, physics, and engineering. Here are a few examples:
- Geometry: Circles are fundamental shapes in geometry and are used in various geometric calculations and constructions.
- Physics: Circles are used to model circular motion, like the movement of planets around the sun or the path of a spinning object.
- Engineering: Circular shapes are ubiquitous in engineering designs, from gears and wheels to pipes and tanks.
Conclusion
The equation (x+3)^2 + (y-7)^2 = 100 provides a concise way to represent a circle with a specific center and radius. By understanding the standard form and its components, we can analyze and visualize the circle, and apply its principles in various fields.