%5E2%2B(y-3)%5E2%3D49.jpeg "(x+1)^2+(y-3)^2=49")
Unveiling the Secrets of (x+1)^2 + (y-3)^2 = 49
The equation (x+1)^2 + (y-3)^2 = 49 represents a circle in the Cartesian coordinate system. Let's dive into its properties and understand how to interpret this equation.
The Standard Equation of a Circle
The general standard form for the equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
Where:
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
Analyzing our Equation
Comparing our equation (x+1)^2 + (y-3)^2 = 49 to the standard form, we can identify the following:
- Center: (h, k) = (-1, 3)
- Radius: r^2 = 49 => r = 7
Visualizing the Circle
With the center and radius in hand, we can easily visualize the circle on a graph:
- Plot the center: Locate the point (-1, 3) on the coordinate plane.
- Draw the circle: Starting from the center, draw a circle with a radius of 7 units.
Key Points to Remember
- The equation (x+1)^2 + (y-3)^2 = 49 describes a circle with its center at (-1, 3) and radius of 7 units.
- The standard form of the circle equation is a powerful tool for understanding and graphing circles.
- Understanding the relationship between the equation and its corresponding geometric shape allows for deeper insights into mathematical concepts.
By analyzing the equation (x+1)^2 + (y-3)^2 = 49, we have unveiled the hidden properties of this circle. This journey helps us appreciate the connection between algebraic equations and geometric figures.