91Ó°ÊÓ

Find the length of the parametric curve defined over the given interval. $$ x=\cos t, y=\ln (\sec t+\tan t)-\sin t ; 0 \leq t \leq \frac{\pi}{4} $$

Find a Cartesian equation of the graph of $$ r^{2}-6 r(\cos \theta+\sin \theta)+9=0 $$ and then sketch the graph.

How long is the latus rectum (chord through the focus perpendicular to the major axis) for the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1 ?\)

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. $$ 4 x^{2}+x y+4 y^{2}=56 $$

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. $$ 4 x y-3 y^{2}=64 $$

Find a Cartesian equation of the graph of \(r^{2} \cos 2 \theta=9\) and then sketch the graph.

In Problems \(46-49\), use a computer or graphing calculator to graph the given equation. Make sure that you choose a sufficiently large interval for the parameter so that the entire curve is drawn. $$ r=\sqrt{1-0.5 \sin ^{2} \theta} $$

Find the length of the parametric curve defined over the given interval. $$ x=\sin t-t \cos t, y=\cos t+t \sin t ; \frac{\pi}{4} \leq t \leq \frac{\pi}{2} $$

Use a computer or graphing calculator to graph the given equation. Make sure that you choose a sufficiently large interval for the parameter so that the entire curve is drawn. $$ r=\cos (13 \theta / 5) $$

Find the slope of the tangent line to the graph of \(r=3+3 \cos \theta\) at the point on the graph where \(\theta=\frac{1}{6} \pi .\)