91Ó°ÊÓ

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. \(r=\frac{3}{4-8 \cos \theta}\)

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. \(r=\frac{4}{2+3 \cos \theta}\)

Find the vertices and foci of the ellipse and sketch its graph. $$ x^{2}+3 y^{2}+2 x-12 y+10=0 $$

Identify the curve by finding a Cartesian equation for the curve. \(r=4 \sec \theta\)

Find \(d y / d x\) and \(d^{2} y / d x^{2} .\) For which values of \(t\) is the curve concave upward? $$ x=\cos t, \quad y=\sin 2 t, \quad 0

Graph the curve and find the area that it encloses. $$ r=1+5 \sin 6 \theta $$

(a) Eliminate the parameter to ind a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. $$ x=\sqrt{t+1}, \quad y=\sqrt{t-1} $$

(a) Find the eccentricity and directrix of the conic \(r=1 /(1-2 \sin \theta)\) and graph the conic and its directrix. (b) If this conic is rotated counterclockwise about the origin through an angle \(3 \pi / 4,\) write the resulting equation and graph its curve.

\(17-20\) Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. $$ x=t^{3}-3 t, \quad y=t^{2}-3 $$

(a) Eliminate the parameter to ind a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. $$ x=\sinh t, \quad y=\cosh t $$