Introduction

Understanding the relationship between the equation of a circle and its radius is fundamental in geometry. The standard form of a circle's equation, (x^2 y^2 r^2), directly connects the coordinates to the circle's properties. This article will explore how to find the radius of a circle given by the equation (x^2 y^2 8).

Standard Form of a Circle's Equation

The standard form of a circle's equation with the center at the origin (0,0) is given by [x^2 y^2 r^2,] where (r) is the radius of the circle. This equation represents all points in the plane that are a distance (r) from the center (0,0).

Given Equation and Radius Calculation

The equation provided is (x^2 y^2 8). By comparing this to the standard form, we see that (r^2 8). To find the radius (r), we take the square root of both sides:

[r sqrt{8} 2sqrt{2}.] Thus, the radius of the circle is (2sqrt{2}).

Discussion on Related Equations and Consequences

Comparing with (x^2 y^2 8):

(x^2 y^2 8) directly translates to (r^2 8), and hence (r 2sqrt{2}).

Radius Calculation with Squaring:

If we consider multiplying 8 by 2 to give (8^2 64), this would be incorrect for the original equation, as it changes the significance of the term from (r^2) to (r^4).

Origin as Center:

The equation (x^2 y^2 8) describes a circle with its center at the origin (0,0) and a radius of (2sqrt{2}).

Specific Points on the Circle:

On the (x)-axis (where (y 0)), the equation simplifies to (x^2 8). Solving for (x), we get (x pm 2sqrt{2}). This means the circle passes through the points ((2sqrt{2}, 0)) and ((-2sqrt{2}, 0)).

Verification with Distance Formula:

Using the coordinates (0,0) and (2sqrt{2}, 0), the distance (radius) can be calculated using the distance formula:

[r sqrt{(2sqrt{2} - 0)^2 (0 - 0)^2} sqrt{8} 2sqrt{2}.]

General Case for Circle's Radius

The general form of a circle's equation is (x^2 y^2 r^2), where (r) is the radius. Given an equation in this form, the radius can be directly determined:

[r sqrt{r^2}.] For example, in the equation (x^2 y^2 16), the radius would be (r sqrt{16} 4).

Conclusion

The radius of the circle given by the equation (x^2 y^2 8) is (2sqrt{2}). This value can be derived from the standard form of the circle's equation, and its correctness can be verified by considering specific points on the circle and using the distance formula. Understanding and applying these principles is crucial for solving related geometric problems.