Question

A family of curves called hyperbolas (discussed in Section 4) has the parametric equations \(x=a\) tan \(t, y=b \sec t\) for \(-\pi

Short answer

Question: Graph the hyperbola given by the parametric equations \(x=\tan t\) and \(y=\sec t\), for \(-\pi<t<\pi\) and \(|t|\neq\pi/2\). Answer: The graph is a hyperbola with foci at \((-1, 0)\) and \((1, 0)\), and asymptotes given by \(y=\pm x\). The curve is generated in the direction from the upper branch to the lower branch, passing through the foci, as \(t\) increases from \(-\pi\) to \(\pi\).

Step-by-step solution

Write down the parametric equations and substitute values of a and b

Given parametric equations are \(x=a\tan t\) and \(y=b\sec t\). Here, \(a=b=1\), so: \(x=\tan t\) \(y=\sec t\)

Rewrite the equations in terms of \(\cos t\) and \(\sin t\)

Since \(\tan t=\frac{\sin t}{\cos t}\) and \(\sec t=\frac{1}{\cos t}\), we can rewrite our equations as: \(x=\frac{\sin t}{\cos t}\) \(y=\frac{1}{\cos t}\)

Write down the equation for the hyperbola in Cartesian coordinates

From the equations in step 2, we can write \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\): \(\sin t = x\cos t\) Cosine is given by the second equation: \(\cos t = \frac{1}{y}\) Now, we can substitute this value of \(\cos t\) to get sine: \(\sin t = x\left(\frac{1}{y}\right)\) \(\sin t = \frac{x}{y}\) Using the Pythagorean identity \(\sin^2 t+\cos^2 t=1\), we can obtain the Cartesian equation: \(\left(\frac{x}{y}\right)^2+\left(\frac{1}{y}\right)^2=1\) \(x^2+y^2=y^2\)

Graph the hyperbola using the derived equation

The equation in Cartesian coordinate system is \(x^2+y^2=y^2\). This is a standard hyperbola which has the following form \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\). It has foci at \((\pm ab, 0)\), so foci at \((\pm 1, 0)\), the left focus is at \((-1, 0)\) and the right focus is at \((1, 0)\). It also has asymptotes \(y=\pm\frac{b}{a}x\) or in this case \(y=\pm x\). To graph the hyperbola, draw the asymptotes, foci and sketch the curve passing through the foci. The resulting graph will show two branches of the hyperbola.

Indicate direction in which the curve is generated as \(t\) increases from \(t=-\pi\) to \(t=\pi\)

Keep in mind that \(|t|\neq\pi/2\). To determine the direction of the generating curve, calculate the coordinates for \(t\) values between the given range and observe how they change as \(t\) increases. For example: When \(t=-\pi\), \(x=\tan -\pi=0\) and \(y=\sec(-\pi)=1\). As \(t\) increases, the value of \(x\) goes from 0 to negative infinity, and \(y\) goes from infinity to negative infinity. When \(t=\pi\), \(x=\tan \pi=0\) and \(y=\sec(\pi)=1\). As \(t\) decreases, the value of \(x\) goes from 0 to positive infinity, and \(y\) goes from infinity to negative infinity. Therefore, as \(t\) increases from \(t=-\pi\) to \(t=\pi\), the curve is generated in the direction from the upper branch towards the lower branch, passing through the foci.

Key concepts

Parametric Equations

Parametric equations are a powerful tool in mathematics, especially when describing curves like hyperbolas. These equations express the coordinates of points on the curve as functions of a variable, often called the parameter. For the hyperbola described in the exercise, the parametric equations are given as \( x = a\tan t \) and \( y = b\sec t \). Here, \( t \) is the parameter, and it varies over a range to generate the curve.

Why use parametric equations?

• They simplify the representation of complex curves that might be difficult to express with a single equation in \( x \) and \( y \).
• They allow for the easy calculation of points along a curve by simply plugging in different \( t \) values. This is particularly useful in computer graphics and animations.
• In the case of the hyperbola, using parametric equations helps highlight how the curve extends to infinity as \( t \) approaches certain critical values, such as \( \pm \pi/2 \).

By substituting different values for \( t \), parametric equations can show how a curve is traversed over a specified range of the parameter.

Conic Sections

Conic sections are the curves obtained by slicing a cone with a plane. These curves include circles, ellipses, parabolas, and hyperbolas. Hyperbolas, specifically, are formed when a plane intersects both nappes of the double cone but not through the vertex. This creates two separate open curves which are mirror images of each other.

Key attributes of hyperbolas:

• They consist of two branches, which open either horizontally or vertically depending on the orientation of the slices.
• The center of a hyperbola is the midpoint of the line segment joining its vertices, and the standard hyperbola is symmetric with respect to both axes.
• The asymptotes of a hyperbola are straight lines that the curve approaches but never touches. For the standard hyperbola with horizontal branches, they are given by the equations \( y = \pm \frac{b}{a}x \).

A hyperbola's equation in Cartesian form, like \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), highlights the differences in its structure from other conic sections. As depicted in the exercise, this algebraic equation is derived from manipulating parametric forms.

Cartesian Coordinates

Cartesian coordinates are a fundamental component in geometry that aid in describing the location of points in a plane. Using an \( x, y \) system, any point can be represented as an ordered pair \( (x, y) \).

Transforming parametric forms into Cartesian coordinates is crucial for visualizing the curve in the familiar rectangular coordinate system. For the given hyperbola, starting with parametric equations \( x = \tan t \) and \( y = \sec t \), the task is to combine them using trigonometric identities to yield a Cartesian form.

Steps to transform include:

• Use fundamental trigonometric identities like \( \tan t = \frac{\sin t}{\cos t} \) and \( \sec t = \frac{1}{\cos t} \).
• Relate the sine and cosine to \( x \) and \( y \) using the parametric equations.
• Simplify to find an expression solely in terms of \( x \) and \( y \), demonstrating their relationship directly without involving the parameter \( t \).

This conversion is not just an algebraic exercise but a crucial step in understanding how the curve manifests in familiar coordinates, thus aiding graphing and further analysis.