Ellipse Exploration Center, Vertices, Eccentricity, Foci, And Axes Lengths For 9x² + 5y² - 30y = 0

by ADMIN 99 views

In the fascinating world of conic sections, the ellipse holds a special place with its elegant shape and intriguing properties. Understanding the ellipse requires us to delve into its key characteristics, such as the center, vertices, foci, eccentricity, and the lengths of its major and minor axes. In this comprehensive guide, we will embark on a step-by-step journey to dissect the equation 9x² + 5y² - 30y = 0, unlocking its secrets and revealing the ellipse's fundamental attributes.

Transforming the Equation: Completing the Square

Our quest begins with the given equation: 9x² + 5y² - 30y = 0. To extract the ellipse's hidden parameters, we must first transform this equation into its standard form. The standard form of an ellipse equation provides a clear representation of its center, axes, and orientation. The key technique we employ here is completing the square, a powerful algebraic method that allows us to rewrite quadratic expressions in a more revealing form.

Let's start by grouping the 'y' terms together: 9x² + (5y² - 30y) = 0. Our focus now shifts to the expression within the parentheses. To complete the square for the 'y' terms, we need to add and subtract a specific constant. This constant is determined by taking half of the coefficient of the 'y' term (-30), squaring it, and multiplying by the coefficient of the y² term (5). In this case, half of -30 is -15, squaring it gives 225, and multiplying by 1/5 (to get the coefficient to be 1 for completing the square) gives 25. Thus, we add and subtract 25 inside the parentheses:

9x² + 5(y² - 6y + 9 - 9) = 0

Notice that we factored out the coefficient '5' from the 'y' terms to ensure the coefficient of y² inside the parentheses is 1, a crucial step for completing the square. Now, we can rewrite the expression inside the parentheses as a perfect square:

9x² + 5((y - 3)² - 9) = 0

Next, distribute the 5 and move the constant term to the right side of the equation:

9x² + 5(y - 3)² - 45 = 0

9x² + 5(y - 3)² = 45

Finally, divide both sides by 45 to obtain the standard form of the ellipse equation:

x²/5 + (y - 3)²/9 = 1

This is the standard form of the ellipse equation, and it holds the key to unlocking the ellipse's secrets.

Decoding the Standard Form: Center, Axes, and Orientation

Now that we have the equation in its standard form, x²/5 + (y - 3)²/9 = 1, we can readily identify the ellipse's center, the lengths of its semi-major and semi-minor axes, and its orientation. The standard form of an ellipse equation centered at (h, k) is:

(x - h)²/a² + (y - k)²/b² = 1 (for a horizontal ellipse, where a > b)

(x - h)²/b² + (y - k)²/a² = 1 (for a vertical ellipse, where a > b)

Comparing our equation with the standard form, we can immediately deduce the following:

  • Center (h, k): The center of the ellipse is at (0, 3). This is the point around which the ellipse is symmetrically arranged.
  • Semi-major axis (a): The larger denominator is 9, so a² = 9, which means a = 3. This is the distance from the center to the vertices along the major axis. The major axis is always the longer axis of the ellipse.
  • Semi-minor axis (b): The smaller denominator is 5, so b² = 5, which means b = √5. This is the distance from the center to the co-vertices along the minor axis. The minor axis is always the shorter axis of the ellipse.
  • Orientation: Since the larger denominator is under the (y - 3)² term, the ellipse has a vertical orientation. This means the major axis is aligned vertically.

Locating the Vertices and Co-vertices

The vertices of an ellipse are the endpoints of its major axis, while the co-vertices are the endpoints of its minor axis. Knowing the center and the lengths of the semi-major and semi-minor axes, we can easily determine the coordinates of these points.

For our ellipse, which has a vertical orientation:

  • Vertices: Since the major axis is vertical and a = 3, the vertices are located 3 units above and below the center. Thus, the vertices are at (0, 3 + 3) = (0, 6) and (0, 3 - 3) = (0, 0).
  • Co-vertices: Since the minor axis is horizontal and b = √5, the co-vertices are located √5 units to the left and right of the center. Thus, the co-vertices are at (0 + √5, 3) = (√5, 3) and (0 - √5, 3) = (-√5, 3).

Unveiling the Foci: The Ellipse's Focal Points

The foci (plural of focus) are two special points within the ellipse that play a crucial role in its definition. An ellipse can be defined as the set of all points such that the sum of the distances from any point on the ellipse to the two foci is constant. The foci always lie on the major axis, inside the ellipse. To find the foci, we need to calculate the distance 'c' from the center to each focus. This distance is related to the semi-major axis 'a' and the semi-minor axis 'b' by the following equation:

c² = a² - b²

In our case, a² = 9 and b² = 5, so:

c² = 9 - 5 = 4

Taking the square root of both sides, we get c = 2. Since the ellipse is vertically oriented, the foci are located 2 units above and below the center. Therefore, the foci are at:

  • (0, 3 + 2) = (0, 5)
  • (0, 3 - 2) = (0, 1)

Eccentricity: Measuring the Ellipse's Shape

Eccentricity is a dimensionless parameter that quantifies how much an ellipse deviates from a perfect circle. It is denoted by 'e' and is defined as the ratio of the distance from the center to a focus (c) to the length of the semi-major axis (a):

e = c/a

The eccentricity of an ellipse always lies between 0 and 1. An eccentricity of 0 corresponds to a circle, while an eccentricity closer to 1 indicates a more elongated ellipse. For our ellipse, c = 2 and a = 3, so the eccentricity is:

e = 2/3

This value indicates that our ellipse is moderately elongated, not too circular and not too stretched out.

Summarizing the Ellipse's Characteristics

To consolidate our findings, let's summarize the key characteristics of the ellipse defined by the equation 9x² + 5y² - 30y = 0:

  • Center: (0, 3)
  • Vertices: (0, 6) and (0, 0)
  • Co-vertices: (√5, 3) and (-√5, 3)
  • Foci: (0, 5) and (0, 1)
  • Semi-major axis (a): 3
  • Semi-minor axis (b): √5
  • Eccentricity: 2/3
  • Length of major axis: 2a = 6
  • Length of minor axis: 2b = 2√5

Conclusion: Mastering the Ellipse

In this comprehensive exploration, we have successfully dissected the equation 9x² + 5y² - 30y = 0 and unveiled the ellipse's essential characteristics. By mastering the techniques of completing the square, understanding the standard form of the ellipse equation, and applying the relevant formulas, we have determined the center, vertices, foci, eccentricity, and the lengths of the axes. This knowledge empowers us to fully grasp the geometry and properties of ellipses, making them less of a mystery and more of a familiar shape in the world of conic sections.

This detailed analysis not only provides a complete understanding of this specific ellipse but also equips you with the tools to tackle similar problems involving ellipses and other conic sections. The journey into the world of ellipses is a rewarding one, filled with mathematical elegance and practical applications in various fields, from astronomy to engineering.