91Ó°ÊÓ

The Cartesian coordinates of a point are given. (i) Find polar coordinates \((r, \theta)\) of the point, where \(r>0\) and \(0 \leqslant \theta<2 \pi\). (ii) Find polar coordinates \((r, \theta)\) of the point, where \(r<0\) and \(0 \leqslant \theta<2 \pi\) (a) \((\sqrt{3},-1)\) (b) \((-6,0)\)

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $$ x=e^{t} \sin \pi t, \quad y=e^{2 t} ; \quad t=0 $$

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$ (y-2)^{2}=2 x+1 $$

Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity 0.6, directrix \(r=4 \csc \theta\)

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(r \geqslant 1\)

\(7-8\) Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter. $$ x=1+\ln t, \quad y=t^{2}+2 ; \quad(1,3) $$

(a) 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. (b) Eliminate the parameter to ind a Cartesian equation of the curve. $$ x=t^{2}-3, \quad y=t+2, \quad-3 \leq t \leq 3 $$

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$ y^{2}+6 y+2 x+1=0 $$

(a) 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. (b) Eliminate the parameter to ind a Cartesian equation of the curve. $$ x=\sin t, \quad y=1-\cos t, \quad 0 \leqslant t \leqslant 2 \pi $$

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$ 2 x^{2}-16 x-3 y+38=0 $$