Equation Of Circle Q Centered At Origin Radius 9 Units

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In the realm of geometry, circles hold a fundamental position, characterized by their elegant symmetry and consistent properties. Understanding the equation of a circle is crucial for various mathematical and real-world applications. This article delves into the specifics of determining the equation of a circle, particularly focusing on circle Q, which is centered at the origin and has a radius of 9 units. We will explore the standard form of a circle's equation and apply it to identify the correct representation for circle Q. Additionally, we will dissect each of the provided options, explaining why some are incorrect and highlighting the precise equation that fits the given criteria. This comprehensive guide aims to clarify any confusion and solidify your understanding of circle equations.

The foundation for identifying the equation of circle Q lies in understanding the standard form of a circle's equation. In coordinate geometry, a circle can be uniquely defined by its center and radius. The standard equation that encapsulates this information is expressed as:

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

Where:

  • (h, k) represents the coordinates of the center of the circle.
  • r denotes the radius of the circle.

This equation is derived from the Pythagorean theorem and represents all points (x, y) that are a distance r away from the center (h, k). The values of h, k, and r directly determine the circle's position and size in the coordinate plane. When the circle is centered at the origin, which is the case for circle Q, the coordinates of the center are (0, 0). This simplifies the standard equation, making it easier to identify and apply. Understanding this fundamental form is essential for analyzing and manipulating circle equations in various mathematical contexts.

When a circle is centered at the origin, its equation simplifies significantly. The origin, represented by the coordinates (0, 0), becomes the center of the circle. Substituting h = 0 and k = 0 into the standard equation of a circle, (x - h)^2 + (y - k)^2 = r^2, we obtain:

(x - 0)^2 + (y - 0)^2 = r^2

This further simplifies to:

x^2 + y^2 = r^2

This simplified equation is a direct representation of a circle centered at the origin, where 'r' still denotes the radius of the circle. This form is particularly useful because it directly relates the coordinates (x, y) of any point on the circle to the radius. For circle Q, which is centered at the origin and has a radius of 9 units, we can directly apply this simplified equation. By substituting the radius value, we can determine the specific equation that represents circle Q. This understanding of circles centered at the origin is a foundational concept in coordinate geometry, making it easier to visualize and analyze such circles.

Circle Q is defined as having a radius of 9 units. To determine the equation that represents circle Q, we substitute this radius value into the simplified equation for a circle centered at the origin, which is x^2 + y^2 = r^2. Replacing 'r' with 9, we get:

x^2 + y^2 = 9^2

Calculating 9 squared (9^2) gives us 81. Therefore, the equation for circle Q is:

x^2 + y^2 = 81

This equation signifies that any point (x, y) on circle Q is 9 units away from the origin. It's a concise representation of circle Q's properties, encapsulating both its center (at the origin) and its radius (9 units). This application of the radius to the standard equation clearly demonstrates how the geometric properties of a circle directly translate into its algebraic representation. The equation x^2 + y^2 = 81 is the unique equation that accurately describes circle Q in the coordinate plane.

To confirm that x^2 + y^2 = 81 is indeed the correct equation for circle Q, let's analyze the options provided and understand why the others are incorrect:

  • x^2 + y^2 = 3: This equation represents a circle centered at the origin, but with a radius of √3 (since r^2 = 3, r = √3). This does not match the radius of 9 units for circle Q.
  • x^2 + y^2 = 9: This equation also represents a circle centered at the origin, but with a radius of 3 (since r^2 = 9, r = 3). Again, this radius does not match the given radius of 9 units for circle Q.
  • x^2 + y^2 = 81: As derived earlier, this equation perfectly represents circle Q, which is centered at the origin and has a radius of 9 units.
  • (x - 9)^2 + (y - 9)^2 = 1: This equation represents a circle with a center at (9, 9) and a radius of 1 (since r^2 = 1, r = 1). The center is not at the origin, and the radius is incorrect, thus it does not represent circle Q.

By systematically analyzing each option, we can clearly see why x^2 + y^2 = 81 is the only equation that accurately describes circle Q. The other options either have the wrong radius or are not centered at the origin.

In conclusion, the correct equation that represents circle Q, which is centered at the origin and has a radius of 9 units, is x^2 + y^2 = 81. This equation is derived from the standard form of a circle's equation and accurately reflects the given properties of circle Q. Understanding the relationship between a circle's geometric properties and its algebraic representation is crucial in coordinate geometry. By analyzing the standard equation of a circle, particularly when centered at the origin, we can easily determine the correct equation given the radius. This comprehensive analysis not only confirms the correct answer but also provides a clear understanding of why the other options are incorrect, reinforcing the foundational principles of circle equations in geometry.

The final answer is x^2 + y^2 = 81.