Ellipse Equation Solver: Find Vertices, Foci & More

by Felix Dubois 52 views

Hey guys! Today, we're diving deep into the fascinating world of ellipses. We've got a juicy equation to dissect: 25x² + 16y² + 100x - 96y - 156 = 0. Our mission? To transform this seemingly complex expression into a beautiful ellipse, pinpointing its key features – the vertices, foci, latus rectum, eccentricity, and directrices – and of course, sketching its elegant form. Buckle up, because we're about to embark on an elliptical adventure!

1. Taming the Equation: Completing the Square

So, how do we even begin to decipher this equation? The magic lies in a technique called completing the square. This nifty trick allows us to rewrite the equation in a standard form that reveals the ellipse's secrets. Let's break it down step by step:

  1. Grouping Like Terms: First, we gather the x-terms and y-terms together, moving the constant to the other side: (25x² + 100x) + (16y² - 96y) = 156
  2. Factoring Out Leading Coefficients: Next, we factor out the coefficients of the squared terms. This is crucial for completing the square: 25(x² + 4x) + 16(y² - 6y) = 156
  3. Completing the Square (x-terms): Now comes the heart of the method. To complete the square for the x-terms, we take half of the coefficient of the x-term (which is 4), square it (4/2 = 2, 2² = 4), and add it inside the parenthesis. But remember, we're actually adding 25 * 4 to the left side, so we must add the same to the right side to maintain balance: 25(x² + 4x + 4) + 16(y² - 6y) = 156 + 25 * 4
  4. Completing the Square (y-terms): We repeat the process for the y-terms. Half of -6 is -3, and (-3)² = 9. We add 9 inside the parenthesis, which means we're adding 16 * 9 to the left side. So, we add the same to the right side: 25(x² + 4x + 4) + 16(y² - 6y + 9) = 156 + 100 + 16 * 9
  5. Rewriting as Squared Terms: Now, we can rewrite the expressions in parentheses as squared terms. This is the payoff for all our hard work! 25(x + 2)² + 16(y - 3)² = 156 + 100 + 144
  6. Simplifying: Let's simplify the right side: 25(x + 2)² + 16(y - 3)² = 400
  7. Standard Form: Finally, we divide both sides by 400 to get the equation in the standard form of an ellipse: (x + 2)² / 16 + (y - 3)² / 25 = 1

Boom! We've transformed the original equation into a much friendlier form. This standard form is our key to unlocking the ellipse's properties. From here, we can easily identify the center, semi-major axis, and semi-minor axis, which are the building blocks for finding everything else.

2. Decoding the Ellipse: Center, Axes, and Orientation

Now that we have the equation in standard form, (x + 2)² / 16 + (y - 3)² / 25 = 1, it's time to extract the vital statistics of our ellipse. This equation is like a treasure map, guiding us to the ellipse's center, its major and minor axes, and its overall orientation.

  • Center: The center of the ellipse is the point (h, k) in the standard equation. In our case, we have (x + 2) and (y - 3), which translates to a center at (-2, 3). Think of this as the heart of the ellipse, the point around which everything is symmetrical.
  • Semi-major and Semi-minor Axes: The denominators under the squared terms hold the key to the ellipse's shape. The larger denominator corresponds to the semi-major axis (a), and the smaller one corresponds to the semi-minor axis (b). Here, we have 16 and 25. Since 25 is larger, we have:
    • a² = 25, so a = 5 (semi-major axis)
    • b² = 16, so b = 4 (semi-minor axis)
  • Orientation: The position of the larger denominator tells us whether the ellipse is vertically or horizontally oriented. Since the larger denominator (25) is under the (y - 3)² term, this ellipse is vertically oriented. This means the major axis is vertical, and the ellipse is stretched along the y-axis.

With this information in hand, we've laid the foundation for finding the vertices, foci, and other essential features of our ellipse. Understanding the center, axes, and orientation is like having the blueprint for our elliptical structure. We're well on our way to fully understanding this fascinating shape.

3. Pinpointing the Key Features: Vertices, Foci, and Eccentricity

With the basics covered, we're ready to pinpoint the key features that define our ellipse: the vertices, foci, and eccentricity. These elements paint a complete picture of the ellipse's shape and position in the coordinate plane.

  • Vertices: The vertices are the endpoints of the major axis. Since our ellipse is vertically oriented, the major axis runs along the y-axis. The vertices are located 'a' units above and below the center. We know a = 5, and the center is (-2, 3). Therefore, the vertices are:

    • Vertex 1: (-2, 3 + 5) = (-2, 8)
    • Vertex 2: (-2, 3 - 5) = (-2, -2)

    These points mark the extreme ends of the ellipse along its major axis.

  • Foci: The foci (plural of focus) are two special points inside the ellipse that play a crucial role in its definition. The distance from any point on the ellipse to the two foci always adds up to the same constant value. To find the foci, we first need to calculate the distance 'c' from the center to each focus using the formula: c² = a² - b² In our case, a² = 25 and b² = 16, so: c² = 25 - 16 = 9 c = √9 = 3

    Since the ellipse is vertical, the foci are located 'c' units above and below the center:

    • Focus 1: (-2, 3 + 3) = (-2, 6)
    • Focus 2: (-2, 3 - 3) = (-2, 0)

    The foci are key to understanding the ellipse's curvature and its unique reflective properties.

  • Eccentricity: Eccentricity (e) is a measure of how