91Ó°ÊÓ

Find polar coordinates that satisfy (a) \(r>0,-\pi<\theta \leq \pi\) (b) \(r<0,-\pi<\theta \leq \pi\) for each point with the given rectangular coordinates. $$ (1,2) $$

Identify the given rotated conic. Find the polar coordinates of its vertex or vertices. $$ r=\frac{4}{1+\cos (\theta-\pi / 4)} $$

Find polar coordinates that satisfy (a) \(r>0,-\pi<\theta \leq \pi\) (b) \(r<0,-\pi<\theta \leq \pi\) for each point with the given rectangular coordinates. $$ (-3,4) $$

Find the \(x\) - and \(y\) -intercepts of the given curves. $$ x=-1+2 \cos t, y=1+2 \sin t, 0 \leq t \leq 2 \pi $$

In Problems 31 and 32 , the graph of the given equation is a spiral. Sketch its graph. $$ r=2^{\theta}, \theta \geq 0 \text { (logarithmic) } $$

Identify the given rotated conic. Find the polar coordinates of its vertex or vertices. $$ r=\frac{4}{1+\cos (\theta-\pi / 4)} $$

The graph of the given equation is a spiral. Sketch its graph. $$ r \theta=\pi, \theta>0 \text { (hyperbolic) } $$

Find polar coordinates that satisfy (a) \(r>0,-\pi<\theta \leq \pi\) (b) \(r<0,-\pi<\theta \leq \pi\) for each point with the given rectangular coordinates. $$ (1,-1) $$

Find the \(x\) - and \(y\) -intercepts of the given curves. $$ x=1+\sin t, y=\sin t-\cos t, 0 \leq t \leq 2 \pi $$

Show that parametric equations for a line through \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) are $$ x=x_{1}+\left(x_{2}-x_{1}\right) t, \quad y=y_{1}+\left(y_{2}-y_{1}\right) t, \quad-\infty