91Ó°ÊÓ

Find a parametrization for the curve. the ray (half line) with initial point (2,3) that passes through the point (-1,-1)

Sketch the region defined by the inequalities \(-1 \leq r \leq 2\) and \(-\pi / 2 \leq \theta \leq \pi / 2\)

Graph the sets of points whose polar coordinates satisfy the equations and inequalities. $$-\pi / 2 \leq \theta \leq \pi / 2, \quad 1 \leq r \leq 2$$

Give information about the foci and vertices of ellipses centered at the origin of the \(x y\) -plane. In each case, find the ellipse's standard-form equation from the given information. $$\text { Foci: }(\pm \sqrt{2}, 0) \quad \text { Vertices: }(\pm 2,0)$$

Find a parametrization for the curve. the ray (half line) with initial point (-1,2) that passes through the point (0, 0)

Give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the \(x y\) -plane. In each case, find the hyperbola's standard-form equation in Cartesian coordinates. Eccentricity:\(\quad\) \(2\) Vertices: \(\quad(\pm 2,0)\)

Find the lengths of the curves. The parabolic segment \(r=2 /(1-\cos \theta), \quad \pi / 2 \leq \theta \leq \pi\)

Find the lengths of the curves. $$x=t^{3}, \quad y=3 t^{2} / 2, \quad 0 \leq t \leq \sqrt{3}$$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities. $$0 \leq \theta \leq \pi / 2, \quad 1 \leq|r| \leq 2$$

Give information about the foci and vertices of ellipses centered at the origin of the \(x y\) -plane. In each case, find the ellipse's standard-form equation from the given information. $$\text { Foci: }(0,\pm 4) \quad \text { Vertices: }(0,\pm 5)$$