Find The Center Of The Ellipse: 4 X 2 + 9 Y 2 + 8 X − 36 Y + 4 = 0 4x^2 + 9y^2 + 8x - 36y + 4 = 0 4 X 2 + 9 Y 2 + 8 X − 36 Y + 4 = 0 A. Center: (2, 1) B. Center: (1, 2) C. Center: (-1/3, 1) D. Center: (-1, 2)

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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 standard form of the equation of an ellipse with its center at (h, k) is given by:

(x−h)2/a2+(y−k)2/b2=1{(x - h)^2/a^2 + (y - k)^2/b^2 = 1}

where a and b are the semi-major and semi-minor axes of the ellipse, respectively. 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:

Ax2+Bxy+Cy2+Dx+Ey+F=0{Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0}

where A, B, C, D, E, and F are constants. To find the center of the ellipse, we need to rewrite the equation in standard form.

Step 1: Group the Terms

The first step is to group the terms involving x and y. We can rewrite the equation as:

4x2+8x+9y2−36y+4=0{4x^2 + 8x + 9y^2 - 36y + 4 = 0}

Step 2: Complete the Square

Next, we need to complete the square for both x and y terms. To do this, we will add and subtract the square of half the coefficient of x and y to the equation.

For the x terms, we have:

4(x2+2x)+9y2−36y+4=0{4(x^2 + 2x) + 9y^2 - 36y + 4 = 0}

We can rewrite the x term as:

4(x2+2x+1)−4+9y2−36y+4=0{4(x^2 + 2x + 1) - 4 + 9y^2 - 36y + 4 = 0}

Simplifying the equation, we get:

4(x+1)2+9y2−36y−4=0{4(x + 1)^2 + 9y^2 - 36y - 4 = 0}

Similarly, for the y terms, we have:

4(x+1)2+9(y2−4y)−4=0{4(x + 1)^2 + 9(y^2 - 4y) - 4 = 0}

We can rewrite the y term as:

4(x+1)2+9(y2−4y+4)−36−4=0{4(x + 1)^2 + 9(y^2 - 4y + 4) - 36 - 4 = 0}

Simplifying the equation, we get:

4(x+1)2+9(y−2)2−40=0{4(x + 1)^2 + 9(y - 2)^2 - 40 = 0}

Step 3: Rewrite the Equation in Standard Form

Now that we have completed the square, we can rewrite the equation in standard form.

4(x+1)2+9(y−2)2=40{4(x + 1)^2 + 9(y - 2)^2 = 40}

Dividing both sides by 40, we get:

(x+1)2/10+(y−2)2/4.4=1{(x + 1)^2/10 + (y - 2)^2/4.4 = 1}

Finding the Center

The center of the ellipse is given by the values of h and k in the standard form of the equation. In this case, we have:

h=−1{h = -1}

k=2{k = 2}

Therefore, the center of the ellipse is (-1, 2).

Conclusion

In this article, we have explored how to find the center of an ellipse given its equation in general form. We have used the steps of grouping the terms, completing the square, and rewriting the equation in standard form to find the center of the ellipse. The center of the ellipse is given by the values of h and k in the standard form of the equation.

Answer

The correct answer is:

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

A: The general form of an ellipse equation is given by:

Ax2+Bxy+Cy2+Dx+Ey+F=0{Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0}

where A, B, C, D, E, and F are constants.

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 follow these steps:

  1. Group the terms involving x and y.
  2. Complete the square for both x and y terms.
  3. Rewrite the equation in standard form.

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

A: The standard form of an ellipse equation is given by:

(x−h)2/a2+(y−k)2/b2=1{(x - h)^2/a^2 + (y - k)^2/b^2 = 1}

where a and b are the semi-major and semi-minor axes of the ellipse, respectively.

Q: How do I find the values of h and k in the standard form of the equation?

A: To find the values of h and k in the standard form of the equation, you need to look at the equation and identify the values that are being subtracted from x and y. These values are the coordinates of the center of the ellipse.

Q: What is the significance of the center of an ellipse?

A: The center of an ellipse is an important concept in mathematics and has many real-world applications. For example, in physics, the center of an ellipse can represent the position of a planet or a satellite in orbit around a star or a moon. In engineering, the center of an ellipse can be used to design and optimize the shape of a structure or a system.

Q: Can I find the center of an ellipse using a calculator or a computer program?

A: Yes, you can find the center of an ellipse using a calculator or a computer program. Many calculators and computer programs have built-in functions and tools that can help you find the center of an ellipse given its equation in general form.

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 values of h and k correctly
  • Not checking the equation for errors or inconsistencies

Q: Can I find the center of an ellipse if the equation is not in general form?

A: Yes, you can find the center of an ellipse even if the equation is not in general form. However, you may need to use other methods or techniques to find the center, such as using the equation of the ellipse in a different form or using a different approach to solve the problem.

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

A: Some real-world applications of finding the center of an ellipse include:

  • Designing and optimizing the shape of a structure or a system
  • Modeling and analyzing the motion of a planet or a satellite
  • Creating and optimizing the shape of a curve or a surface
  • Solving problems in physics, engineering, and other fields that involve ellipses and their properties.