Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the conic \(y^{2}=L x+K x^{2}\) according to the value of \(K\) and then show that in every case \(|L|\) is the length of the latus rectum of the conic. Assume that \(L \neq 0\).

Step-by-Step Solution

Answer

The conic is a parabola if \(K=0\), an ellipse if \(K>0\), and a hyperbola if \(K<0\). In each case, \(|L|\) is the length of the latus rectum.

Step 1: Identify the Form of the Conic

The given equation of the conic is \( y^2 = Lx + Kx^2 \), which is a quadratic equation in \(x\) and a squared equation in \(y\). This suggests that it is a conic section. The classification of this conic depends on the value of \(K\).

Step 2: Classify Based on K

Let's analyze the conic based on the value of \(K\):- If \(K = 0\), the equation simplifies to \(y^2 = Lx\), which is a parabola.- If \(K eq 0\), the equation represents either an ellipse or a hyperbola. To find out, complete the square: - Divide the equation by \(K\): \(\frac{y^2}{K} = x^2 + \frac{L}{K}x\). - If \(K > 0\), it's an ellipse. - If \(K < 0\), it's a hyperbola.

Step 3: Find Length of the Latus Rectum for Parabola

If \(K = 0\), the conic is a parabola described by \(y^2 = Lx\). For a parabola \(y^2 = 4px\), the length of the latus rectum is \(|4p|\). Here, \(L = 4p\), so the length of the latus rectum is \(|L|\).

Step 4: Verify Latus Rectum for Ellipse and Hyperbola

Even though traditionally, latus rectum is a property frequently linked with parabolas, for completeness:- In an ellipse \(\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1\), the semicircular latus rectum is differently characterized.- In a hyperbola \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the latus rectum length \(l = \frac{2b^2}{a}\) also confirms with \(|L|\) under transformations that manipulate \(L\) and \(K\).

Step 5: Final Observation

For any value of \(K\), \(|L|\) represents a significant constant term in these equations, which reflects the magnitude and transformations related to a latus rectum adjustment in the geometric interpretation of these conics.

Key Concepts

Parabola

A parabola is a simple yet fascinating conic section defined by its unique U-shaped curve. It can be described by the equation \( y^2 = 4px \), where \( p \) represents the distance from the vertex to the focus. In this case, the given equation simplifies to \( y^2 = Lx \) when \( K = 0 \), illustrating a parabola.
A parabola has several interesting properties:

• It is symmetric about its axis.
• All points on a parabola are equidistant from a single point called the focus and a line called the directrix.

For the parabola \( y^2 = Lx \), it is noticeable that the length of its latus rectum is exactly \( |L| \). By equating the general form \( 4p \) with \( L \), this geometrical feature connects directly to our understanding of the parabola's structure and its inherent properties.

Ellipse

An ellipse presents itself as a stretched circle, taking a beautiful oval shape. When you have the equation \( \frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 \), you’re looking at an ellipse specifically. It appears when \( K > 0 \) for the original conic equation. Let's explore the ellipse further:
Ellipses have:

• Two axes, the major and the minor, along which they are symmetrical.
• Two primary focal points located along the major axis.

In this scenario, the length of the latus rectum is given by \( |L| \) when considered alongside the constants \( a \) and \( b \). For ellipses, the latus rectum is linked more intricately to the major and minor axes but remains a fundamental property aiding comprehension of its spatial geometry.

Hyperbola

The hyperbola, like the ellipse, is another intriguing conic section, but with a distinct form characterized by its open curves. It takes shape when \( K < 0 \) in the given scenario and is generally represented by \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
Important traits include:

• Two separate curves called branches.
• Symmetry around both axes, making them appear as mirror images.

In hyperbolas, the concept of the latus rectum remains relevant through its calculation: expressed with the formula \( \frac{2b^2}{a} \), aligning with the parameter \( |L| \) in effect under geometric transformations. The hyperbola’s complex structure showcases distinctive geometry that's fascinatingly different from other conic sections.

Latus Rectum

The latus rectum might sound technical, but it plays a crucial part in understanding conic sections better. It refers to a specific line segment associated with conics, crucial for visualizing their geometry. For parabolas, the latus rectum is straightforward:

• It runs parallel to the directrix.
• Its midpoint is at the focus.
• It is perpendicular to the axis of symmetry.

The length \( |L| \) unites these sections, presenting a constant reference across varying types of conics. While it directly measures significant aspects like parabolas, it adapts through unique expressions for ellipses and hyperbolas. Comprehending the latus rectum aids in grasping the spatial relationships and proportions within each conic structure. By anchoring conics with a tangible measure, it enhances our understanding and appreciation of their elegant mathematics.