Determining the Diameter and Radius of a Circle using its Equation

In this article, we will explore the process of determining the diameter and radius of a circle given its equation. Understanding the general form of a circle's equation and how to extract key information from it is crucial for solving such problems. We will walk through an example and discuss the contributions of two individuals, Lucy and Joshua, to determine who is correct.

General Form of a Circle's Equation

The standard form of a circle's equation is given by:

[ (x - h)^2 (y - k)^2 r^2 ]

Here, ((h, k)) represents the center of the circle, and (r) represents the radius. The given equation is in the form of:

[ x^2 - 4^2 y^2 36 ]

This equation is not in the standard form, but we can still determine the radius and diameter from it.

Analysis of the Given Equation

Let's break down the given equation step-by-step:

[ x^2 - 16 y^2 36 ]

We can rewrite this equation to better resemble the general form:

[ (x - 0)^2 (y - 0)^2 36 16 ]

Combining the constants on the right side:

[ (x - 0)^2 (y - 0)^2 52 ]

Although this form is not the standard form, we can still extract the key information. The term (52) is (r^2), so:

[ r^2 52 ]

However, this form is not correct. The correct form should be:

[ x^2 (y - 0)^2 36 16 - 16 ]

Which simplifies to:

[ x^2 y^2 36 ]

Now, the equation is in the standard form, where ((0, 0)) is the center, and (r^2 36).

Calculating the Radius and Diameter

From (r^2 36), we can find the radius (r) by taking the square root of both sides:

[ r sqrt{36} 6 ]

The diameter (d) of a circle is twice the radius:

[ d 2r 2 times 6 12 ]

Conclusion: Lucy is Correct

Based on the analysis, Lucy is correct. The diameter of the circle is 12 units, not 72 units as claimed by Joshua.

Summary:

Radius: 6 units Diameter: 12 units

To further clarify, let's consider the comments provided by Alice and Juwan. Alice correctly notes that the radius is 6 units, while Juwan incorrectly states that the radius is 36 units. Both Alice and Lucy are correct in their analysis.

Summary:

Alice and Lucy are correct. Juwan is incorrect.

Additional Insights

In another example, Joe and Alice provide additional information to solve the equation correctly. Alice and Juwan's discussions reinforce the importance of correctly interpreting the equation and applying the correct mathematical principles.

For instance, if the equation is given as:

[ x - 4^2 y^2 36 ]

We can rewrite it as:

[ x^2 - 16 y^2 36 ]

Which simplifies to:

[ x^2 y^2 52 ]

Here, the radius is:

[ r sqrt{52} approx 7.21 ]

And the diameter is:

[ d 2 times 7.21 14.42 ]

This example further illustrates the importance of correctly transforming the equation into the standard form before calculating the radius and diameter.