91Ó°ÊÓ

Write a polar equation of a conic with the focus at the origin and the given data. Parabola, directrix \(x=-3\)

Find the area of the region that is bounded by the given curve and lies in the specified sector. $$ r=\cos \theta, \quad 0 \leqslant \theta \leqslant \pi / 6 $$

Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity 1.5, directrix \(y=2\)

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$ 2 x=-y^{2} $$

Find the area of the region that is bounded by the given curve and lies in the specified sector. $$ r=\sin \theta+\cos \theta, \quad 0 \leqslant \theta \leqslant \pi $$

\(3-6\) Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $$ x=t^{3}+1, \quad y=t^{4}+t ; \quad t=-1 $$

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. $$ x=t+\sin t, \quad y=\cos t, \quad-\pi \leqslant t \leqslant \pi $$

Plot the point whose polar coordinates are given. Then ind the Cartesian coordinates of the point. (a) \((2,3 \pi / 2)\) (b) \((\sqrt{2}, \pi / 4)\) (c) \((-1,-\pi / 6)\)

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. $$ x=e^{-t}+t, \quad y=e^{t}-t, \quad-2 \leqslant t \leqslant 2 $$

Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity \(3, \quad\) directrix \(x=3\)