%5E2%2B(y-3)%5E2%3D169.jpeg "(x+5)^2+(y-3)^2=169")
Understanding the Equation (x+5)^2 + (y-3)^2 = 169
The equation (x+5)^2 + (y-3)^2 = 169 represents a circle in the coordinate plane. Let's break down why and explore its key features:
The Standard Equation of a Circle
The general form of the equation for 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.
Applying the Equation to Our Example
In our equation, (x+5)^2 + (y-3)^2 = 169:
- Center: Since the equation is in the form (x - h)^2 + (y - k)^2, we can see that the center of the circle is at (-5, 3). This is because (x + 5) is equivalent to (x - (-5)).
- Radius: The right-hand side of the equation, 169, represents r^2. Therefore, the radius of the circle is √169 = 13.
Key Features and Interpretation
The equation tells us the following about the circle:
- Location: The circle is centered at the point (-5, 3).
- Size: The circle has a radius of 13 units. This means every point on the circle is exactly 13 units away from the center (-5, 3).
Visualizing the Circle
To visualize the circle, you can:
- Plot the center: Locate the point (-5, 3) on a coordinate plane.
- Draw the radius: From the center, draw a line segment of length 13 units in any direction.
- Sketch the circle: Draw a smooth curve around the center, maintaining a constant distance of 13 units from the center.
By understanding the equation and its components, you can easily determine the center, radius, and therefore the shape and position of the circle represented by the equation (x+5)^2 + (y-3)^2 = 169.