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Chapter 18: Problem 13

Find the values of \(k\) for which the intersection of the plane \(x+k y=1\) and the elliptic hyperboloid of two sheets \(y^{2}-x^{2}-z^{2}=1\) is (a) an ellipse and (b) a hyperbola.

Short Answer

Expert verified
For an ellipse: -1 < k < 1 For a hyperbola: k < -1 or k > 1

Step by step solution

01

- Understand the problem

Determine the values of the parameter k such that the intersection of the plane and the hyperboloid is either an ellipse or a hyperbola.
02

- Substitute the plane into the hyperboloid

Replace x in the hyperboloid equation using the plane equation, i.e., substitute x = 1 - ky into the hyperboloid equation.
03

- Form the substituted equation

The modified hyperboloid equation becomes: y^2 - (1 - ky)^2 - z^2 = 1.
04

- Simplify the equation

Expand and simplify the equation: y^2 - (1 - 2ky + k^2 y^2) - z^2 = 1 y^2 - 1 + 2ky - k^2 y^2 - z^2 = 1 Combine like terms: (1 - k^2)y^2 + 2ky - z^2 - 2 = 0.
05

- Determine the discriminant condition

For the equation to represent an ellipse, the coefficient of y^2 term should be positive, for hyperbola, it should be negative.
06

- Condition for ellipse

Ellipse occurs when the coefficient of y^2 is positive: 1 - k^2 > 0 k^2 < 1 -1 < k < 1.
07

- Condition for hyperbola

Hyperbola occurs when the coefficient of y^2 is negative: 1 - k^2 < 0 k^2 > 1 k < -1 or k > 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ellipse Condition
To understand when the intersection of a plane and an elliptic hyperboloid is an ellipse, we need to look at the condition related to the equation coefficients. The original hyperboloid equation after substitution and simplification is:
  • (1 - k^2)y^2 + 2ky - z^2 - 2 = 0.
An ellipse is achieved if the coefficient of the squared term of y,
  • (1 - k^2),
is positive. This positive condition translates to
  • 1 - k^2 > 0.
Solving this inequality, we get:
  • -1 < k < 1.
Hence, the values of k that satisfy this condition will make the intersection an ellipse.
Hyperbola Condition
Similarly, to find out when the intersection is a hyperbola, we need to focus on the same equation. This time, we need the coefficient of the squared term of y to be negative. For this to be true,
  • (1 - k^2)
must be less than zero. The inequality becomes:
  • 1 - k^2 < 0.
Solving this, we get:
  • k^2 > 1
which means that
  • k < -1 or k > 1.
These values of k indicate the points where the intersection will be a hyperbola.
Parameter Determination
Determining the parameter k is crucial for identifying the type of conic section formed by the intersection. First, we substitute the plane equation into the hyperboloid equation. Next, we solve for the values of k that meet the conditions for hyperbola and ellipse. These criteria involve checking the coefficients in the simplified equation:
  • (1 - k^2)y^2 + 2ky - z^2 - 2 = 0.
By analyzing if
  • 1 - k^2 > 0
for an ellipse, or
  • 1 - k^2 < 0
for a hyperbola,
  • We determine the required range of values for k.
In summary:
  • For an ellipse:
    • -1 < k < 1.
  • For a hyperbola:
    • k < -1 or k > 1.
Equation Substitution
Substituting one equation into another is a powerful method to simplify complex problems. For this exercise, we start by substituting the plane equation,
  • x + ky = 1,
into the hyperboloid equation,
  • y^2 - x^2 - z^2 = 1.
We express x from the plane equation as
  • x = 1 - ky
and replace x in the hyperboloid equation:
  • y^2 - (1 - ky)^2 - z^2 = 1.
Simplifying further, we get:
  • y^2 - (1 - 2ky + k^2 y^2) - z^2 = 1.
Expanding and combining like terms results in:
  • (1 - k^2)y^2 + 2ky - z^2 - 2 = 0.
This new equation helps us analyze the nature of the conic section formed by the intersection, making it an essential part of the solution.

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