Find The Center Of This Ellipse. X 2 + 4 Y 2 − 10 X − 40 Y + 121 = 0 X^2 + 4y^2 - 10x - 40y + 121 = 0 X 2 + 4 Y 2 − 10 X − 40 Y + 121 = 0 Center: ([?], [?])

Mar 13, 2025 by ADMIN 157 views

**Finding the Center of an Ellipse: A Step-by-Step Guide**

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Introduction

In mathematics, an ellipse is a closed curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant. The equation of an ellipse in standard form is given by $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , where $(h,k)$ represents the center of the ellipse. However, not all ellipse equations are in standard form. In this article, we will explore how to find the center of an ellipse given its equation in general form.

General Form of an Ellipse Equation

The general form of an ellipse equation is given by $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ . To find the center of the ellipse, we need to rewrite the equation in standard form. The given equation is $x^2 + 4y^2 - 10x - 40y + 121 = 0$ . Our goal is to rewrite this equation in standard form and identify the center of the ellipse.

Completing the Square

To rewrite the equation in standard form, we need to complete the square for both the $x$ and $y$ terms. Completing the square involves adding and subtracting a constant term to create a perfect square trinomial. For the $x$ term, we need to add and subtract $(10/2)^2 = 25$ inside the parentheses. For the $y$ term, we need to add and subtract $(40/2)^2 = 400$ inside the parentheses.

import sympy as sp
x, y = sp.symbols('x y')

eq = x2 + 4*y2 - 10x - 40y + 121

eq_x = (x - 5)**2 - 25

eq_y = 4*(y - 10)**2 - 400

eq_std = eq_x + eq_y

Rewriting the Equation in Standard Form

Now that we have completed the square for both the $x$ and $y$ terms, we can rewrite the equation in standard form.

# Rewrite the equation in standard form
eq_std = (x - 5)**2 + 4*(y - 10)**2 - 625 - 400 + 121

Identifying the Center of the Ellipse

The standard form of the ellipse equation is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ . Comparing this with our rewritten equation, we can identify the center of the ellipse as $(h,k) = (5,10)$ .

Conclusion

In this article, we have explored how to find the center of an ellipse given its equation in general form. We completed the square for both the $x$ and $y$ terms and rewrote the equation in standard form. By comparing the rewritten equation with the standard form of the ellipse equation, we identified the center of the ellipse as $(5,10)$ . This method can be applied to any ellipse equation in general form to find its center.

Example Problems

Problem 1

Find the center of the ellipse given by the equation $x^2 + 2y^2 - 6x - 12y + 36 = 0$ .

Solution

To find the center of the ellipse, we need to complete the square for both the $x$ and $y$ terms.

# Define the variables
x, y = sp.symbols('x y')

eq = x2 + 2*y2 - 6x - 12y + 36

eq_x = (x - 3)**2 - 9

eq_y = 2*(y - 6)**2 - 72

eq_std = eq_x + eq_y

The center of the ellipse is identified as $(h,k) = (3,6)$ .

Problem 2

Find the center of the ellipse given by the equation $4x^2 + y^2 + 8x - 2y + 4 = 0$ .

Solution

To find the center of the ellipse, we need to complete the square for both the $x$ and $y$ terms.

# Define the variables
x, y = sp.symbols('x y')

eq = 4x2 + y2 + 8x - 2*y + 4

eq_x = 4*(x + 1)**2 - 16

eq_y = (y - 1)**2 - 1

eq_std = eq_x + eq_y

The center of the ellipse is identified as $(h,k) = (-1,1)$ .

Final Thoughts

Finding the center of an ellipse is an essential skill in mathematics, particularly in algebra and geometry. By completing the square for both the $x$ and $y$ terms and rewriting the equation in standard form, we can identify the center of the ellipse. This method can be applied to any ellipse equation in general form to find its center. With practice and patience, you can master this skill and become proficient in finding the center of ellipses.

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Q: What is the general form of an ellipse equation?

A: The general form of an ellipse equation is given by $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ .

Q: How do I find the center of an ellipse given its equation in general form?

A: To find the center of an ellipse given its equation in general form, you need to complete the square for both the $x$ and $y$ terms and rewrite the equation in standard form. The standard form of the ellipse equation is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , where $(h,k)$ represents the center of the ellipse.

Q: What is completing the square?

A: Completing the square involves adding and subtracting a constant term to create a perfect square trinomial. For the $x$ term, you need to add and subtract $(D/2)^2$ inside the parentheses. For the $y$ term, you need to add and subtract $(E/2)^2$ inside the parentheses.

Q: How do I complete the square for the x term?

A: To complete the square for the $x$ term, you need to add and subtract $(D/2)^2$ inside the parentheses. For example, if the equation is $x^2 + Dx + E = 0$ , you would add and subtract $(D/2)^2$ to get $(x + D/2)^2 - (D/2)^2 + E = 0$ .

Q: How do I complete the square for the y term?

A: To complete the square for the $y$ term, you need to add and subtract $(E/2)^2$ inside the parentheses. For example, if the equation is $y^2 + Ey + F = 0$ , you would add and subtract $(E/2)^2$ to get $(y + E/2)^2 - (E/2)^2 + F = 0$ .

Q: What is the standard form of the ellipse equation?

A: The standard form of the ellipse equation is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , where $(h,k)$ represents the center of the ellipse.

Q: How do I identify the center of the ellipse from the standard form of the equation?

A: To identify the center of the ellipse from the standard form of the equation, you need to look at the values of $h$ and $k$ in the equation. The center of the ellipse is given by $(h,k)$ .

Q: Can I use this method to find the center of any ellipse equation in general form?

A: Yes, you can use this method to find the center of any ellipse equation in general form. However, you need to make sure that the equation is in the correct form and that you complete the square correctly.

Q: What are some common mistakes to avoid when finding the center of an ellipse?

A: Some common mistakes to avoid when finding the center of an ellipse include:

Not completing the square correctly
Not rewriting the equation in standard form
Not identifying the center of the ellipse correctly
Not checking the equation for any errors or inconsistencies

Q: How can I practice finding the center of an ellipse?

A: You can practice finding the center of an ellipse by working through example problems and exercises. You can also try finding the center of different types of ellipse equations, such as those with different values of $A$ , $B$ , $C$ , $D$ , and $E$ .

Q: What are some real-world applications of finding the center of an ellipse?

A: Finding the center of an ellipse has many real-world applications, including:

Designing and building elliptical shapes, such as those used in architecture and engineering
Analyzing and modeling the motion of objects in elliptical orbits
Understanding and predicting the behavior of elliptical systems, such as those found in physics and astronomy

Q: Can I use this method to find the center of other types of conic sections, such as parabolas and hyperbolas?

A: No, this method is specifically designed for finding the center of ellipses. However, you can use similar methods to find the center of other types of conic sections, such as parabolas and hyperbolas.

Q: What are some additional resources for learning about finding the center of an ellipse?

A: Some additional resources for learning about finding the center of an ellipse include:

Textbooks and online resources on algebra and geometry
Online tutorials and videos on finding the center of an ellipse
Practice problems and exercises on finding the center of an ellipse
Real-world examples and applications of finding the center of an ellipse