Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ \frac{-x^{2}}{9}+\frac{y}{4}=0 $$

Step-by-Step Solution

Answer

The conic is a vertical parabola.

Step 1: Rewrite the equation

The given equation is \(\frac{-x^{2}}{9} + \frac{y}{4} = 0\). Start by isolating one of the variables. Let's move \(\frac{y}{4}\) to the other side: \(\frac{-x^{2}}{9} = -\frac{y}{4}\).

Step 2: Simplify the equation

To simplify, multiply through by -1 to get positive terms: \(\frac{x^2}{9} = \frac{y}{4}\). Multiply both sides by 36 (the least common multiple of 9 and 4) to eliminate the fractions: \(36 \cdot \frac{x^2}{9} = 36 \cdot \frac{y}{4}\). This simplifies to \(4x^2 = 9y\).

Step 3: Standard form of a parabola

The equation \(4x^2 = 9y\) resembles the standard form of a parabola with a vertical axis. An equation of the form \((ax^2 = by)\) typically represents a vertical parabola. This means the conic is a vertical parabola.

Key Concepts

Parabolas

Parabolas are a type of conic section that you can find in many real-world applications, such as satellite dishes and car headlights. They have a distinct U-shape and can either open upwards, downwards, or sideways.

Parabolas have some unique characteristics that make them stand out:

• They have a vertex, which is the highest or lowest point on the curve, depending on its orientation.
• The line through the vertex is called the axis of symmetry, which means the parabola is symmetrical along this line.
• Parabolas have a focus and a directrix; the distance from any point on the parabola to these is always equal. This property defines the curve.

Understanding these features helps in identifying and working with these curves. For example, in the exercise provided, the equation \(4x^2 = 9y\) is typical of a vertical parabola, indicating that the parabolic opening is aligned with the y-axis.

Equation of a Parabola

The equation of a parabola can be represented in different standard forms depending on its orientation. The two most common forms are for parabolas that open vertically and horizontally.

• A vertical parabola typically has an equation of the form \(x^2 = 4py\), where \(p\) defines the distance from the vertex to the focus or directrix. It opens upwards if \(p > 0\) and downwards if \(p < 0\).
• A horizontal parabola, on the other hand, is in the form \(y^2 = 4px\). Here, it opens to the right if \(p > 0\) and to the left if \(p < 0\).

In the solution, isolating and simplifying the variables led to the equation \(4x^2 = 9y\). This is akin to the vertical parabola form, confirming that the parabola's axis is vertical. Recognizing these forms quickly allows you to determine important details about the parabola, such as the direction it opens and where its axis of symmetry lies.

Conic Equation Classification

Conic sections, named so because they are the curves obtained by intersecting a plane with a cone, include circles, ellipses, parabolas, and hyperbolas. Each type is defined by its own unique equation.

For classification purposes, a general second-degree equation is often used: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). The values of the coefficients \(A, B,\) and \(C\) are crucial in determining which conic section is represented:

• If \(B^2 - 4AC < 0\), it suggests a circle or ellipse.
• If \(B^2 - 4AC = 0\), it is a parabola, as seen in the exercise problem after transformation.
• If \(B^2 - 4AC > 0\), it signifies a hyperbola.

Breaking down an equation like the one given allows for precise identification by comparing its structure against these criteria. Our example finally fits the parabola classification through simplification, confirming the absence of an \(xy\) term and ensuring a unique geometric interpretation.