Finding The Center Of An Ellipse $x^2+36 Y^2-16 X-72 Y+64=0$

Do you find yourself staring at the equation of an ellipse, feeling lost and unsure how to pinpoint its center? You're not alone! Ellipses, with their elongated circular shape, might seem intimidating at first. However, with a systematic approach and a bit of algebraic manipulation, finding the center of an ellipse becomes a straightforward task. This comprehensive guide will walk you through the process, breaking down each step with clear explanations and examples. We will focus on the given equation: $x^2 + 36y^2 - 16x - 72y + 64 = 0$ and demonstrate how to transform it into a standard form that readily reveals the center's coordinates. By the end of this article, you'll be equipped with the knowledge and skills to confidently tackle any ellipse equation and locate its center.

The center of an ellipse is a fundamental property that defines its position in the coordinate plane. It is the midpoint of the major and minor axes, the two lines of symmetry that characterize the ellipse's shape. Knowing the center is crucial for understanding the ellipse's overall geometry and its relationship to other geometric figures. Furthermore, the center serves as a reference point for determining other key features, such as the foci, vertices, and the lengths of the axes. In various applications, from physics to engineering, accurately locating the center of an ellipse is essential for modeling and analyzing elliptical phenomena. For instance, in astronomy, the orbits of planets around the sun are elliptical, and knowing the center of the orbit helps predict the planet's position over time. Similarly, in optics, the design of elliptical mirrors and lenses relies on the precise location of the ellipse's center to focus light effectively. Therefore, mastering the technique of finding the center of an ellipse is not only a valuable mathematical skill but also a practical tool for solving real-world problems.

Understanding the Standard Form of an Ellipse

The standard form of an ellipse's equation is the key to unlocking its center coordinates. This form provides a clear and concise representation of the ellipse's properties, making it easy to identify the center, axes lengths, and orientation. There are two standard forms, depending on whether the major axis is horizontal or vertical:

• Horizontal Major Axis: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
• Vertical Major Axis: $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$

In both forms, (h, k) represents the coordinates of the center. The values 'a' and 'b' are the semi-major and semi-minor axes, respectively. The semi-major axis ('a') is half the length of the major axis, while the semi-minor axis ('b') is half the length of the minor axis. The major axis is the longer axis of the ellipse, and the minor axis is the shorter axis. If a > b, the major axis is horizontal in the first equation and vertical in the second equation. Conversely, if b > a, the major axis is vertical in the first equation and horizontal in the second equation. Understanding these standard forms is crucial because once the equation of an ellipse is transformed into this form, the center (h, k) can be directly read off. The process of transforming a general equation into the standard form involves completing the square, a technique that will be discussed in detail in the next section. By recognizing the standard form and its components, you gain a powerful tool for analyzing and understanding ellipses.

Completing the Square: The Key Technique

Completing the square is a fundamental algebraic technique used to transform quadratic expressions into a perfect square trinomial. This technique is essential for rewriting the given equation of the ellipse into its standard form, allowing us to easily identify the center. Let's break down the process step-by-step, focusing on the equation $x^2 + 36y^2 - 16x - 72y + 64 = 0$. The first step is to group the x-terms and y-terms together and move the constant term to the right side of the equation:

(x² - 16x) + (36y² - 72y) = -64

Next, factor out the coefficient of the squared term for both x and y. In this case, we factor out 36 from the y-terms:

(x² - 16x) + 36(y² - 2y) = -64

Now, we complete the square for the x-terms and the y-terms separately. To complete the square for a quadratic expression of the form x² + bx, we take half of the coefficient of the x-term (which is b/2), square it ((b/2)²), and add it to both sides of the equation. For the x-terms, the coefficient of x is -16. Half of -16 is -8, and squaring -8 gives us 64. So, we add 64 inside the parenthesis for the x-terms. For the y-terms, the coefficient of y is -2. Half of -2 is -1, and squaring -1 gives us 1. So, we add 1 inside the parenthesis for the y-terms. However, since we factored out 36 from the y-terms, we are actually adding 36 * 1 = 36 to the left side of the equation. Therefore, we must also add 36 to the right side. The equation now becomes:

(x² - 16x + 64) + 36(y² - 2y + 1) = -64 + 64 + 36

Now, we can rewrite the expressions in parentheses as perfect squares:

(x - 8)² + 36(y - 1)² = 36

This step is the culmination of completing the square. The expressions (x - 8)² and (y - 1)² are perfect squares, making the equation much easier to manipulate into the standard form of an ellipse. In the next section, we will further transform this equation into the standard form by dividing both sides by 36.

Transforming to Standard Form and Identifying the Center

After completing the square, we have the equation (x - 8)² + 36(y - 1)² = 36. To transform this equation into the standard form of an ellipse, we need to divide both sides by 36 to make the right side equal to 1:

$\frac{(x - 8)²}{36} + \frac{36(y - 1)²}{36} = \frac{36}{36}$

Simplifying, we get:

$\frac{(x - 8)²}{36} + \frac{(y - 1)²}{1} = 1$

Now, the equation is in the standard form: $\frac{(x - h)²}{a²} + \frac{(y - k)²}{b²} = 1$. Comparing this to our equation, we can identify the values of h, k, a², and b².

• (x - h)² = (x - 8)², so h = 8
• (y - k)² = (y - 1)², so k = 1
• a² = 36, so a = 6
• b² = 1, so b = 1

Therefore, the center of the ellipse is (h, k) = (8, 1). The semi-major axis is a = 6, and the semi-minor axis is b = 1. Since a² is under the (x - 8)² term, the major axis is horizontal. This means the ellipse is stretched horizontally more than it is stretched vertically. The center, (8, 1), is the midpoint of this ellipse, and all other features of the ellipse, such as its vertices and foci, are referenced from this point. By successfully transforming the given equation into standard form, we have clearly identified the center of the ellipse, which is the key to understanding its geometry and position in the coordinate plane.

Conclusion: You've Found the Center!

Congratulations! By following the steps outlined in this guide, you have successfully found the center of the ellipse defined by the equation $x^2 + 36y^2 - 16x - 72y + 64 = 0$. The center is located at the point (8, 1). This process involved completing the square, transforming the equation into standard form, and then identifying the center coordinates (h, k). You now possess a valuable skill for analyzing and understanding ellipses. Remember, the center is a fundamental property of an ellipse, serving as a reference point for determining other key features and understanding its overall geometry.

Understanding ellipses and their properties has wide-ranging applications in various fields. From the elliptical orbits of planets in astronomy to the design of optical systems in physics and engineering, ellipses play a crucial role in modeling and analyzing real-world phenomena. The ability to find the center of an ellipse, as you've learned in this guide, is a fundamental step in understanding and working with these applications. By mastering this technique, you've not only enhanced your mathematical skills but also opened doors to exploring the fascinating world of elliptical geometry and its practical uses. Keep practicing and applying this knowledge, and you'll become even more confident in your ability to tackle ellipse-related problems. The journey of mathematical discovery is continuous, and with each new concept you master, you expand your understanding of the world around you.