Question

Explorations Hyperbolas Let \(x=a \sec t\) and \(y=b \tan t\) (a) Writing to Learn Let \(a=1,2,\) or \(3, b=1,2,\) or \(3,\) and graph using the parameter interval \((-\pi / 2, \pi / 2)\) . Explain what you see, and describe the role of \(a\) and \(b\) in these parametric equations. (Caution: If you get what appear to be asymptomes, try using the approximation \([-1.57,1.57]\) for the parameter interval.) (b) Let \(a=2, b=3,\) and graph in the parameter interval \((\pi / 2,3 \pi / 2)\) . Explain what you see. (c) Writing to Learn Let \(a=2, b=3,\) and graph using the parameter interval \((-\pi / 2,3 \pi / 2) .\) Explain why you must be careful about graphing in this interval or any interval that contains \(\pm \pi / 2\) . (d) Use algebra to explain why \(\left(\frac{x}{a}\right)^{2}-\left(\frac{y}{b}\right)^{2}=1\) (e) Let \(x=a\) tan \(t\) and \(y=b\) sec \(t .\) Repeat (a), (b), and (d) using an appropriate version of \((\mathrm{d}) .\)

Short answer

The parameters \(a\) and \(b\) affect the width and height of the hyperbolas respectively. Changing the interval reveals a different part of the hyperbolas. The equation of hyperbola can be acquired from the parametric equations.

Step-by-step solution

Graphing the Parametric Equations

Plot the parametric equations \(x=a \sec t\) and \(y=b \tan t\) with \(a=1,2,3\) and \(b=1,2,3\) respectively. Use the parameter interval \((-\pi / 2, \pi / 2)\) for each set of \(a\) and \(b\). These plots will form the shape of hyperbolas.

Analyze the Graphs

Analyze the obtained graphs to observe the role of \(a\) and \(b\) in the parametric equations. By comparing graph patterns, we notice that the values of \(a\) and \(b\) affect the width and height of the hyperbolas respectively.

Graph with New Values for a and b

Change the values of \(a\) and \(b\) to \(a=2, b=3\). Now graph the parametric equations in the parameter interval \((\pi / 2,3 \pi / 2)\). This graph reveals another part of the hyperbolas.

Algebraic Reasoning

For these parametric equations, if we square and subtract them, we obtain the equation of a hyperbola: \(\left(\frac{x}{a}\right)^{2}-\left(\frac{y}{b}\right)^{2}=1\). Squaring each equation results in \(\frac{x^{2}}{a^{2}}=cosec^{2}t\) and \(\frac{y^{2}}{b^{2}}=tan^{2}t\), where subtraction yields the stated equation.

Repeat Steps with Altered Parametric Formulas

Repeat the first three steps with the new parametric formulas, \(x=a \tan t\) and \(y=b \sec t\). And for part (d), sqaring the equations, we note that \(\frac{x^{2}}{a^{2}}=tan^{2}t\) and \(\frac{y^{2}}{b^{2}}=cosec^{2}t\). Summing these equations obtains the equation of another form of the hyperbola: \(\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1\).