%5E2%2B(y-3)%5E2%3D25.jpeg "(x-4)^2+(y-3)^2=25")
Exploring the Circle: (x-4)^2 + (y-3)^2 = 25
This equation, (x-4)^2 + (y-3)^2 = 25, represents a circle. Let's break down why and understand its key features.
The Standard Equation of a Circle
The standard equation of a circle with center (h, k) and radius r is: (x - h)^2 + (y - k)^2 = r^2
Analyzing the Equation
Comparing our equation to the standard form, we can identify:
- Center: (h, k) = (4, 3)
- Radius: r^2 = 25, so r = 5
Visualizing the Circle
This information tells us that the circle:
- Is centered at the point (4, 3).
- Has a radius of 5 units.
To visualize it, you can plot the center point on a coordinate plane and draw a circle with a radius of 5 units extending in all directions from the center.
Key Concepts
- Center: The center of a circle is the point equidistant from all points on the circle's circumference.
- Radius: The radius of a circle is the distance from the center to any point on the circle.
Understanding the standard equation and its components allows you to quickly determine the center and radius of any circle represented in this form. This knowledge is crucial for solving various geometric problems and analyzing circular shapes in different contexts.