Problem 34 - Essential Calculus Early Transcendentals

Question

Show that the curve of intersection of the surfaces \(x^{2}+2 y^{2}-z^{2}+3 x=1\) and \(2 x^{2}+4 y^{2}-2 z^{2}-5 y=0\) lies in a plane.

Step-by-Step Solution

Answer

The intersection lies in a plane because it can be expressed linearly in terms of one variable without introducing other dependencies.

Step 1: Identify the Surfaces

The problem provides two surfaces: \( S_1: x^2 + 2y^2 - z^2 + 3x = 1 \) and \( S_2: 2x^2 + 4y^2 - 2z^2 - 5y = 0 \). The task is to show that their intersection is a plane curve.

Step 2: Form Equation for the Curve of Intersection

To find the curve where these surfaces intersect, consider setting the equations of the surfaces equal (since both equal different constants, we can subtract one from the other):\[x^2 + 2y^2 - z^2 + 3x - 1 - (2x^2 + 4y^2 - 2z^2 - 5y) = 0\]Simplify this equation to obtain a new relation.

Step 3: Simplify the Curve Equation

Subtract the second surface \( S_2 \) from the first surface \( S_1 \) to get:\[-x^2 - 2y^2 + z^2 + 3x - 1 - (-2x^2 - 4y^2 + 2z^2 - 5y) = 0\]Simplify:\[x^2 + 2y^2 - z^2 + 3x - 1 + 2x^2 + 4y^2 - 2z^2 - 5y = 0\]\[3x^2 + 6y^2 - 3z^2 + 3x - 5y - 1 = 0\]Divide the entire equation by 3 to simplify:\[x^2 + 2y^2 - z^2 + x - \frac{5}{3} y - \frac{1}{3} = 0\]

Step 4: Show Planarity by Solving for One Variable

In the equation from Step 3, rearrange and solve for \( z^2 \):\[z^2 = x^2 + 2y^2 + x - \frac{5}{3} y - \frac{1}{3}\]Check for constraints that set this equal such that it aligns with a definition of a planar equation (e.g., no dependency results in a linear function of two other variables). Here, the relation is already holding \( z \) as expressed through \( x \) and \( y \), suggesting planar alignment.

Step 5: Verify Independence from Third Variable

Examine both original equations to ensure they provide a consistent and simplifying result when expressed as linear in terms of a third variable. Observe:When one parameter (say \( z \)) can be represented without additional variables or transformed into a function of others without additional dependencies, it refers to a planar relationship.Thus, each original constraint allows translation to linear components when subtracted to baseline curve lying in-plane context.

Key Concepts

Planar Curves

A planar curve is essentially a curve that entirely lies on a single two-dimensional plane. Understanding planar curves involves realizing that the curve itself can be traced out in two dimensions, even though it initially appears within a three-dimensional space.
It's fascinating because you can think of a planar curve as a 'flattened' version of what might have been a more complex three-dimensional interaction, like the intersection of surfaces here.
Let's dive into why a curve is planar in terms of the exercise:

• The curve of intersection between two surfaces means that wherever these surfaces 'meet', forms a line of interaction, which can settle into a plane.
• If the equation derived from these intersections reduces into a form where no new dimensions are introduced, then the entire intersection resides within a plane.
• Based on the simplification process of their intersection, the variables like \( z^2 \) can sometimes be fully described by other variables like \( x \) and \( y \), indicating no third dimension influences the curve.
• The exercise solution specifically shows that \( z \) can be expressed entirely in terms of \( x \) and \( y \), hence the curve is planar.

Coordinate Geometry

Coordinate Geometry, or Cartesian geometry, is the branch of geometry where the position of points on the plane is described using an ordered pair of numbers. It acts as the bridge between algebraic equations and geometric curves.
The goal is to translate equations (like those defining surfaces) into physical shapes and intersecting constructs like lines or curves:

• In coordinate geometry, we use coordinates to represent the shapes of surfaces and their intersections.
• This involves assigning a coordinate system with x, y, and z axes which help visualize these shapes geometrically.
• Mapping our three-dimensional equations into two-dimensional contexts makes it possible to speak about planar curves or surfaces within coordinate spaces.

In our exercise, the manipulation and simplification of the original equations allow us to see the plane materializing from these interferences. The process of subtraction of the equations of the surfaces leverages algebra within this geometric space to show how they exist together.
Thus, coordinate geometry is vital for interpreting complex surfaces and simplifying our understanding of intersections like that of \( S_1 \) with \( S_2 \).

Analytical Geometry

Analytical Geometry, often overlapping with coordinate geometry, emphasizes solving geometrical problems using algebraic equations. It enables the transition between algebraic expressions and geometric figures.
Analytical Geometry in our context focuses on:

• The manipulation of algebraic equations to deduce their geometric implications.
• This is done by aligning them to known forms like lines or planes, making them easier to visualize or prove properties about, such as describing an intersection curve.

Our exercise showcases an excellent application of analytical geometry:

• The equations given govern two distinct surfaces in three-dimensional space. By applying algebraic operations, namely simplifying and subtracting, we deduce a form that centrally describes a plane.
• We use tools of algebra to express one variable seemingly dependent on others, and perceive the plane these surfaces' intersection belongs to as a result.
• This idea highlights how intersections frequently translate to simplified geometric shapes, shown through algebraic form here.

Thus, analytical geometry surrounds the iterative breakdown of equations into a form conveying simple geometric truths, as delineated by a planar intersection.