91Ó°ÊÓ

Find the eccentricity of the ellipse. Then find and graph the ellipse's foci and directrices. $$7 x^{2}+16 y^{2}=112$$

Find the areas of the regions. Inside the oval limaçon \(r=4+2 \cos \theta\)

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$x=3-3 t, \quad y=2 t, \quad 0 \leq t \leq 1$$

Find the eccentricity of the ellipse. Then find and graph the ellipse's foci and directrices. $$2 x^{2}+y^{2}=4$$

Identify the symmetries of the curves. Then sketch the curves in the \(x y\) -plane. $$r=\cos (\theta / 2)$$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities. $$r=2$$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities. $$r \geq 1$$

Find a parametrization for the curve. the lower half of the parabola \(x-1=y^{2}\)

Find the area enclosed by the ellipse $$ x=a \cos t, \quad y=b \sin t, \quad 0 \leq t \leq 2 \pi $$

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