(x-1)^2+(y+3)^2=9

Jasonbradley

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Jun 17, 2024

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Understanding the Equation (x-1)^2 + (y+3)^2 = 9

The equation (x-1)^2 + (y+3)^2 = 9 represents a circle in the Cartesian coordinate system. Let's break down why and explore its key features.

The Standard Form of a Circle

The general 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 (x-1)^2 + (y+3)^2 = 9

Comparing our given equation to the standard form, we can identify:

• Center (h, k): (1, -3)
• Radius r: 3

This means our circle is centered at the point (1, -3) and has a radius of 3 units.

Visualizing the Circle

To visualize the circle, we can follow these steps:

1. Plot the center: Locate the point (1, -3) on the coordinate plane.
2. Mark the radius: From the center, move 3 units to the right, left, up, and down. These points will lie on the circle's circumference.
3. Connect the points: Draw a smooth curve connecting the points you marked. This curve represents the circle.

Key Points

• The equation defines a set of all points that are exactly 3 units away from the point (1, -3).
• Every point on the circle satisfies the equation (x-1)^2 + (y+3)^2 = 9.
• The equation can be used to find the coordinates of points on the circle, and vice versa.

Applications

Circles are fundamental geometric shapes with applications in various fields, including:

• Geometry: Calculating area, circumference, and other properties.
• Physics: Describing the motion of objects in circular paths.
• Engineering: Designing circular structures, gears, and other components.
• Computer Graphics: Creating and manipulating circular objects.

Understanding the equation (x-1)^2 + (y+3)^2 = 9 provides a foundation for exploring the properties and applications of circles in various contexts.