Identifying Conic Sections The Case Of (y+3)² = ⅛(x-5)

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Determining the type of conic section represented by a given equation is a fundamental concept in analytic geometry. In this comprehensive analysis, we will delve into the equation (y+3)² = ⅛(x-5), meticulously dissecting its structure to unveil the specific conic section it describes. Our exploration will not only identify the conic section but also illuminate the underlying principles that govern its classification. This is crucial for anyone studying conic sections, as recognizing the standard forms of these equations allows for quick identification and analysis of their properties.

Deciphering the Equation: Spotting the Parabola

At first glance, the equation (y+3)² = ⅛(x-5) might appear cryptic. However, a closer examination reveals a key characteristic that sets it apart: only one variable is squared. This is the hallmark of a parabola. Unlike ellipses and hyperbolas, which feature both x² and y² terms, a parabola's equation will have either an x² term or a y² term, but not both. This difference in the presence of squared terms is crucial in distinguishing between conic sections. In our equation, the y term is squared, while the x term is linear, immediately pointing us towards a parabolic form. The absence of a squared x term definitively rules out ellipses, hyperbolas, and circles, which all require both variables to be squared.

To further solidify our understanding, let's compare the given equation to the standard form of a parabola. Parabolas can open either horizontally or vertically. The standard form for a parabola opening horizontally is (y - k)² = 4p(x - h), where (h, k) is the vertex of the parabola and p is the distance from the vertex to the focus and from the vertex to the directrix. The equation (y+3)² = ⅛(x-5) closely resembles this standard form, confirming our initial assessment. By rewriting the equation as (y - (-3))² = 4 * (1/32) * (x - 5), we can clearly see the correspondence to the standard form. This comparison allows us to identify the vertex as (5, -3) and the value of p as 1/32, providing a more complete understanding of the parabola's characteristics.

Why Not Hyperbola, Ellipse, or Circle?

To further reinforce our identification of the equation as a parabola, it's essential to understand why it cannot represent the other conic sections: hyperbola, ellipse, or circle. Let's dissect each possibility:

• Hyperbola: Hyperbolas are defined by the difference of the squares of x and y terms. Their standard forms are (x - h)²/a² - (y - k)²/b² = 1 for a horizontal hyperbola and (y - k)²/a² - (x - h)²/b² = 1 for a vertical hyperbola. Notice the crucial minus sign between the squared terms. Our equation (y+3)² = ⅛(x-5) lacks this subtraction and the squared x term, definitively ruling out a hyperbola. The presence of both x² and y² terms, with opposite signs, is a defining feature of hyperbolas, setting them apart from parabolas.
• Ellipse: Ellipses, on the other hand, involve the sum of the squares of x and y terms. Their standard form is (x - h)²/a² + (y - k)²/b² = 1. While an ellipse does have both squared terms, they are added together, not separated by a subtraction sign as in a hyperbola. Again, our equation's lack of a squared x term immediately disqualifies it from being an ellipse. The addition of squared terms, with the same sign, is characteristic of ellipses and distinguishes them from hyperbolas.
• Circle: A circle is a special case of an ellipse where the coefficients of the x² and y² terms are equal. Its standard form is (x - h)² + (y - k)² = r², where r is the radius. Like the ellipse, a circle requires both x and y terms to be squared and added together. Since our equation lacks the x² term, it cannot be a circle. The equal coefficients of the squared terms are a defining feature of circles within the broader category of ellipses.

In summary, the absence of a squared x term and the presence of only a squared y term are definitive indicators that the equation (y+3)² = ⅛(x-5) does not represent a hyperbola, ellipse, or circle. These conic sections all require both variables to be squared, a condition not met by the given equation.

Standard Form Transformation: Unveiling the Parabola's Details

To definitively classify the conic section and extract its key features, transforming the given equation into its standard form is a powerful technique. The equation (y+3)² = ⅛(x-5) is already quite close to the standard form of a horizontal parabola. The standard form equation for a parabola that opens to the right is given by: (y - k)² = 4p(x - h). Here, (h, k) represents the vertex of the parabola, and p is the distance between the vertex and the focus, as well as the distance between the vertex and the directrix.

By comparing (y+3)² = ⅛(x-5) to the standard form, we can identify the key parameters. We can rewrite the equation as (y - (-3))² = ⅛(x - 5). This direct comparison allows us to extract the following information:

• k = -3
• h = 5

The vertex of the parabola is therefore (h, k) = (5, -3). This point represents the "corner" of the parabola and is a crucial element in understanding its position and orientation in the coordinate plane. To find the value of p, we need to equate the coefficient of the (x - h) term in our equation with 4p. In this case, we have 4p = ⅛. Solving for p, we get p = 1/32.

The value of p provides critical information about the parabola's shape and location. Since p is positive, the parabola opens to the right. The focus of the parabola is located at a distance of p units to the right of the vertex, and the directrix is a vertical line located a distance of p units to the left of the vertex. Understanding the standard form not only confirms that the equation represents a parabola but also enables us to determine its vertex, direction of opening, focus, and directrix.

Visualizing the Parabola: Graphing the Equation

A visual representation of the equation (y+3)² = ⅛(x-5) can significantly enhance our understanding of the conic section. Graphing the parabola allows us to see its shape, orientation, and key features in a tangible way. To graph the parabola, we can use the information we've already extracted from the standard form transformation: the vertex (5, -3) and the direction of opening (to the right).

Knowing the vertex is the first step in sketching the parabola. We plot the point (5, -3) on the coordinate plane. Since the parabola opens to the right, we know it will extend horizontally from the vertex. To get a better sense of the parabola's shape, we can find a few additional points on the curve. One way to do this is to choose a few x-values greater than 5 and solve for y. For example, let's choose x = 7:

(y + 3)² = ⅛(7 - 5)

(y + 3)² = ⅛(2)

(y + 3)² = ¼

Taking the square root of both sides, we get:

y + 3 = ±½

Solving for y, we find two points:

• y = -3 + ½ = -2.5
• y = -3 - ½ = -3.5

This gives us two additional points on the parabola: (7, -2.5) and (7, -3.5). By plotting these points and sketching a smooth curve through them and the vertex, we can visualize the parabola. The graph clearly shows the parabolic shape, the vertex at (5, -3), and the opening to the right. Furthermore, the graph visually confirms that this equation does not represent a circle, ellipse, or hyperbola, solidifying our understanding of the equation.

Conclusion: The Parabola Revealed

In conclusion, through careful analysis and transformation of the equation (y+3)² = ⅛(x-5), we have definitively identified it as representing a parabola. The presence of only a squared y term, the transformation into standard form, and the visualization through graphing all confirm this classification. This exercise underscores the importance of understanding the standard forms of conic sections and the unique characteristics that distinguish them. By recognizing the absence of a squared x term and the specific structure of the equation, we were able to confidently rule out hyperbolas, ellipses, and circles. This deep dive into the equation not only answers the initial question but also provides a valuable framework for analyzing and classifying other conic sections. Understanding parabola and other conic sections is fundamental for various fields, including physics, engineering, and computer graphics, making this analysis a crucial stepping stone in mastering analytical geometry.