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Chapter 10: Problem 19

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\tan t, \quad y=\cot t, \quad 0

Short Answer

Expert verified
The curve is a hyperbola given by the equation \(xy=1\) for \(x > 0, y > 0\).

Step by step solution

01

Understand the parametric equations

We have the parametric equations: \( x = \tan t \) and \( y = \cot t \). The parameter \( t \) lies in the interval \( 0 < t < \frac{\pi}{2} \). This means \( \tan t \) and \( \cot t \) are both positive in this interval.
02

Analyze the given interval for t

In the interval \(0 < t < \frac{\pi}{2}\), the function \(\tan t\) increases from 0 to infinity, and \(\cot t\) decreases from infinity to 0.
03

Sketch the curve

Create the xy-plane and plot points for values of \(t\) in the interval. For example, if \(t\) is close to 0, then \(x = \tan t \approx 0\), and \(y = \cot t\) is large. As \(t\) approaches \(\frac{\pi}{2}\), \(x = \tan t\) becomes very large, and \(y = \cot t\) approaches 0. The curve is an asymptotic approach from \((0, \, \infty)\) to \((\infty, \, 0)\).
04

Eliminate the parameter

We use the identities \(x = \tan t\) and \(y = \cot t\). We know that \(\tan t = \frac{1}{\cot t}\), which gives \(x = \frac{1}{y}\). Multiplying both sides by \(y\), we obtain \(xy = 1\).
05

Write the rectangular coordinate equation

Based on the elimination of the parameter, the rectangular-coordinate equation of the curve is \(xy = 1\), which is a standard equation for a hyperbola in the first and third quadrants.

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Key Concepts

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

Rectangular Coordinates
Rectangular coordinates are a way to represent points in a plane using two perpendicular lines, usually labeled as the x-axis and the y-axis. Imagine a grid where each point can be specified by a pair of values \(x, y\). This setup allows us to describe the position of a point through standard horizontal and vertical distance measures from the origin \(0, 0\).

In Cartesian coordinates, the horizontal distance from the y-axis is the x-coordinate, while the vertical distance from the x-axis is the y-coordinate.
  • To move from parametric equations to rectangular coordinates, we use relations between the parametric variables.
  • This transformation typically involves mathematical manipulation to eliminate the parameter and express the relationship exclusively with x and y.
  • In our example, we've changed the parametric forms \(x = \tan t\) and \(y = \cot t\) into a simple equation \(xy = 1\).
Understanding these concepts fully helps navigate between different coordinate systems and enhances problem-solving skills in various mathematical fields.
Eliminating Parameters
Eliminating parameters in mathematics involves removing the independent parameter (in this case, \(t\)) from a system of parametric equations in order to derive a single equation that relates x and y directly. This process is crucial when you aim to express a curve in a standard coordinate system.

In the given exercise, we have \(x = \tan t \) and \(y = \cot t\). By recognizing trigonometric identities, specifically that \(\tan t = \frac{1}{\cot t}\), this relationship can be exploited to eliminate the parameter t. Here's how it looks in steps:
  • Substitute the identity \(\tan t = \frac{1}{\cot t}\) into the equation for x, so you express \(x\) directly in terms of \(y\).
  • Upon multiplying both sides by \(y\), you obtain the rectangular equation \(xy = 1\), which cleanly describes the curve.
This method is very useful, as it allows solving complex systems and sketching curves without referring to the parameter.
Sketching Curves
Sketching curves can initially seem challenging, but it's often a straightforward task with a systematic approach. It involves creating a visual representation of the relationship between variables, in this case, derived from parametric equations.

Let's think about the process for sketching the curve defined by \(x = \tan t\) and \(y = \cot t\).
  • First, identify key points, such as the behavior of the equations at the extremities of the parameter's domain \(0 < t < \frac{\pi}{2}\).
  • At \(t\) close to zero, \(x = \tan t\), very small, while \(y = \cot t\) is very large.
  • Conversely, as \(t\) nears \(\frac{\pi}{2}\), \(x\) becomes large, \(y\) approaches zero.
These points suggest an asymptotic behavior, with the curve stretching from \(0, \infty\) to \(\infty, 0\). Visualizing this, you can sketch a hyperbolic curve that fits the conditions set by the equations. It's crucial to consider how the curve behaves as the parameter changes, gathering insight from the limits of functions involved.

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Most popular questions from this chapter

Find parametric equations for the line with the given properties. Find parametric equations for the circle \(x^{2}+y^{2}=a^{2}\).

A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$r=2^{\theta / 12}, \quad 0 \leq \theta \leq 4 \pi$$

(a)Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$13 x^{2}+6 \sqrt{3} x y+7 y^{2}=16$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\cos t, \quad y=\cos 2 t$$

(a)Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$\sqrt{3} x^{2}+3 x y=3$$
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