Chapter 43: Chapter 43

Draw the graph of the curve whose equation is \(\mathrm{xy}=4\).

Graph the hyperbola \(y^{2}-x^{2}=4\). What are the equations of the asymptotes? Draw the asymptotes.

Determine the intercepts, find the asymptotes, and locate the foci of the following hyperbolas: (a) \(x^{2}-\left(y^{2} / 4\right)=1\). (b) \(\left(y^{2} / 16\right)-\left(x^{2} / 4\right)=1\).

Show informally that \(\mathrm{y}=\pm(\mathrm{b} / \mathrm{a}) \mathrm{x}\) are the equations of the asymptotes of the hyperbola whose equation is $$ \left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1 $$

Consider the equation \(x^{2}-4 y^{2}+4 x+8 y+4=0\) Express this equation in standard form, and determine the center, the vertices, the foci, and the eccentricity of this hyperbola. Describe the fundamental rectangle and find the equations of the 2 asymptotes.

Find the equation of the hyperbola with vertices \(\mathrm{V}_{1}(8,0)\), \(\mathrm{V}_{2}(2,0)\) and eccentricity \(\mathrm{e}=2\)

Show that if the coordinates \((\mathrm{x}, \mathrm{y})\) of a point \(\mathrm{P}\) satisfy $$ \left(x^{2} / 9\right)-\left(y^{2} / 16\right)=1 $$ then \(\left|\mathrm{F}_{1} \mathrm{P}-\mathrm{F}_{2} \mathrm{P}\right|=6\), where \(\mathrm{F}_{1}(-5,0)\) and \(\mathrm{F}_{2}(5,0)\) are the foci.

By definition, if an hyperbola has foci \(F_{1}(-c, 0) F_{2}(c, 0)\), and \(\mathrm{P}(\mathrm{x}, \mathrm{y})\) is a point on the hyperbola, then \(\left|\mathrm{PF}_{1}-\mathrm{PF}_{2}\right|=\mathrm{k}\), where \(\mathrm{k}\) is a constant such that $\mathrm{k}<\mathrm{F}_{1} \mathrm{~F}_{2}=2 \mathrm{c}$ Assuming that the above holds, and defining a such that $\mathrm{a}=\mathrm{K} / 2\(. and a constant \)\mathrm{b}$ such that \(\mathrm{b}^{2}=\mathrm{c}^{2}-\mathrm{a}^{2}\) prove that the equation of the hyperbola is $$ \left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1 $$