Expert Assistance for Parametric equations: Comprehensive Resources and Practice Problems

Replace the Cartesian equation with equivalent polar equations. $x=y$

Do the lines ${\displaystyle \frac{x}{3}=\frac{y}{2}=\frac{z}{2}\text{and}\frac{x}{5}=\frac{y}{3}=\frac{z-4}{2}}$ intersect?

The Cartesian coordinates of a point are given.
(−9, 9)
(i) Find polar coordinates $(r,\theta )$ of the point, where r > 0
and ${\displaystyle 0\le \theta <2?.}$

Consider the parametric equations:
${\displaystyle x=e^t}$
${\displaystyle y=3e^{-2t}}$
Find ${\displaystyle \frac{dy}{dx}}$ at the point $(1,3)$

Replace the Cartesian equation with equivalent polar equations. $xy=2$

Find $y^{\prime }(x)$ and $y^{″}(x)$ of the function $y=f(x)$ defined parametrically as ${\displaystyle x=t^2-3,y=t^3-2t,t=1}$

For the given rectangular equation, write an equivalent polar equation. $x^2+y^2-10x=0$

Find the exact length of the curve. Use a graph to determine the parameter interval.
$r={\cos }^2(\frac{\theta }{2})$

Change from rectangular to spherical coordinates. (Let $\rho \ge 0,0\le \theta \le 2\pi$, and $0\le \varphi \le \pi .)$
a) $(0,-8,0)(\rho ,\theta ,\varphi )=?$
b) $(-1,1,-2)(\rho ,\theta ,\varphi )=?$