Calculating The Radius Of A Circle From Its Equation $x^2 + Y^2 - 2x + 8y - 47 = 0$

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Determining the radius of a circle from its equation is a fundamental concept in analytic geometry. In this comprehensive guide, we will delve deep into the process of finding the radius of a circle given its equation in the general form. Specifically, we will focus on the equation $x^2 + y^2 - 2x + 8y - 47 = 0$. This article aims to provide a clear, step-by-step explanation, ensuring that readers of all backgrounds can grasp the underlying principles and apply them confidently.

Understanding the General Equation of a Circle

At the heart of our quest lies the general equation of a circle. To effectively find the radius, it's crucial to first understand the standard form equation of a circle. The general equation of a circle in the Cartesian coordinate system 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 describes all the points (x, y) that are a distance 'r' away from the center (h, k). The standard form provides a direct way to identify the center and radius, which is essential for our problem-solving approach.

However, the equation we are given, $x^2 + y^2 - 2x + 8y - 47 = 0$, is in a different form – the general form. The general form of a circle's equation is given by:

$$x^2 + y^2 + 2gx + 2fy + c = 0$$

Where:

• The center of the circle is at (-g, -f).
• The radius of the circle, r, can be calculated using the formula: $r = \sqrt{g^2 + f^2 - c}$.

This form is less intuitive at first glance but is mathematically equivalent to the standard form. The key to finding the radius from this general form involves a process called completing the square, which we will explore in detail.

The Method of Completing the Square: A Step-by-Step Guide

To calculate the radius, we'll transform the given equation into the standard form, $x^2 + y^2 - 2x + 8y - 47 = 0$. This transformation is achieved through a technique called completing the square. This method allows us to rewrite quadratic expressions in a more manageable form, revealing the center and radius of the circle. Here’s a step-by-step breakdown of the process:

Step 1: Group Like Terms

Begin by grouping the x terms together and the y terms together. Also, move the constant term to the right side of the equation.

$$(x^2 - 2x) + (y^2 + 8y) = 47$$

This rearrangement sets the stage for completing the square for both the x and y components separately. By isolating the x and y terms, we can focus on transforming each quadratic expression into a perfect square trinomial.

Step 2: Complete the Square for x

To complete the square for the x terms, take half of the coefficient of the x term (-2), square it, and add it to both sides of the equation. Half of -2 is -1, and (-1)^2 is 1.

$$(x^2 - 2x + 1) + (y^2 + 8y) = 47 + 1$$

Adding 1 completes the square for the x terms, creating a perfect square trinomial that can be factored easily. This step is crucial in transforming the equation into a form where the center and radius are readily apparent.

Step 3: Complete the Square for y

Similarly, complete the square for the y terms. Take half of the coefficient of the y term (8), square it, and add it to both sides of the equation. Half of 8 is 4, and 4^2 is 16.

$$(x^2 - 2x + 1) + (y^2 + 8y + 16) = 47 + 1 + 16$$

Adding 16 completes the square for the y terms, mirroring the process we used for the x terms. This step ensures that both the x and y components are in a form that can be easily factored into perfect squares.

Step 4: Factor and Simplify

Now, factor the perfect square trinomials and simplify the right side of the equation.

$$(x - 1)^2 + (y + 4)^2 = 64$$

This step is the culmination of the completing the square process. We have successfully transformed the original equation into the standard form of a circle's equation, which directly reveals the center and radius.

Identifying the Radius

With the equation now in standard form, $(x - 1)^2 + (y + 4)^2 = 64$, we can easily identify the center and the radius. Comparing this with the standard form equation $(x - h)^2 + (y - k)^2 = r^2$, we can see:

• The center of the circle is (h, k) = (1, -4).
• r^2 = 64

To precisely calculate the radius, take the square root of 64:

$$r = \sqrt{64} = 8$$

Therefore, the radius of the circle is 8 units. This result is a direct consequence of the transformation we performed using the completing the square method.

Alternative Method: Using the General Form Formula

As we discussed earlier, the general form of a circle’s equation is $x^2 + y^2 + 2gx + 2fy + c = 0$, and the radius can be found using the formula $r = \sqrt{g^2 + f^2 - c}$. This method provides a direct approach to finding the radius without the need for completing the square.

Step 1: Identify g, f, and c

In our given equation, $x^2 + y^2 - 2x + 8y - 47 = 0$, we can identify the coefficients as follows:

• 2g = -2, so g = -1
• 2f = 8, so f = 4
• c = -47

Step 2: Apply the Formula

Substitute these values into the radius formula:

$$r = \sqrt{g^2 + f^2 - c} = \sqrt{(-1)^2 + (4)^2 - (-47)}$$

Step 3: Calculate the Radius

Simplify the expression:

$$r = \sqrt{1 + 16 + 47} = \sqrt{64} = 8$$

Again, we find that the radius of the circle is 8 units. This method serves as a valuable alternative, confirming our previous result and showcasing the versatility of different approaches in solving mathematical problems.

Conclusion: Mastering Circle Equations

In conclusion, we have successfully determined the radius of the circle given by the equation $x^2 + y^2 - 2x + 8y - 47 = 0$ using two distinct methods: completing the square and applying the general form formula. Both methods have their advantages and provide a robust understanding of circle equations. The radius of the circle is 8 units.

Mastering these techniques not only enhances your problem-solving skills in mathematics but also provides a solid foundation for more advanced concepts in analytic geometry and calculus. Whether you prefer the visual clarity of completing the square or the direct calculation of the formula, the ability to find the radius of a circle from its equation is an invaluable tool.

By understanding the underlying principles and practicing various examples, you can confidently tackle any circle equation and unlock its hidden properties. Remember, the key to success in mathematics lies in a combination of conceptual understanding and diligent practice.