2022-01-17

Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane. (Note dun all four graphs ore symmetric about the lines $y=x$ and $y=-x$.) The hyperbola $xy=4$

nick1337

Expert

Step 1 Since $\mathrm{cot}2\theta =\frac{adj}{opp}$, we let $adj=5$ and $opp=12$. By the Pythegorean Theorem, $\left(hyp{\right)}^{2}=\left(adj{\right)}^{2}+\left(opp{\right)}^{2}$ $\left(hyp{\right)}^{2}={5}^{2}+{12}^{2}$ $\left(hyp{\right)}^{2}=169$ $hyp=±13$ Since $0<\theta <\frac{\pi }{2}$, then $0<2\theta <\pi$. Since $\mathrm{cot}2\theta >0$ in QI, then $\mathrm{cos}2\theta >0$. So, we take $hyp=13$ $\mathrm{cos}2\theta =\frac{adj}{hyp}$ $\mathrm{cos}2\theta =\frac{5}{13}$