(x-4)^2+(y+3)^2=16

3 min read Jun 17, 2024
(x-4)^2+(y+3)^2=16

Understanding the Equation: (x-4)^2 + (y+3)^2 = 16

This equation represents a circle in the Cartesian coordinate system. Let's break down its components to understand why.

The Standard Form of a Circle

The general equation for a circle with center (h, k) and radius r is:

(x - h)^2 + (y - k)^2 = r^2

Identifying the Center and Radius

Comparing our equation, (x - 4)^2 + (y + 3)^2 = 16, with the standard form, we can see:

  • h = 4
  • k = -3
  • r^2 = 16, therefore r = 4

This means:

  • The center of the circle is at the point (4, -3).
  • The radius of the circle is 4 units.

Visualizing the Circle

You can now plot the center (4, -3) on a graph. From this point, move 4 units in all directions (up, down, left, and right) to mark the points that lie on the circle. Connect these points to form a circle.

Key Concepts

  • Circle: A set of all points that are equidistant from a fixed point called the center.
  • Radius: The distance from the center of the circle to any point on the circle.
  • Center: The fixed point from which all points on the circle are equidistant.

Applications

The equation of a circle finds applications in various fields:

  • Geometry: Representing and analyzing circles, calculating their area, circumference, and other geometric properties.
  • Physics: Describing the motion of objects in a circular path, such as planets orbiting a star.
  • Engineering: Designing circular structures, components, and systems.
  • Computer graphics: Creating and manipulating circular objects in digital environments.

Understanding the equation of a circle is fundamental in these and many other areas, allowing us to analyze and solve problems related to circular shapes and their properties.

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