91影视

Identify the conic and calculate the angle of rotation of axes for the curve described by the equation $$ 13 x^{2}-6 \sqrt{3} x y+7 y^{2}-256=0 $$

Plot the point whose polar coordinates are given by first constructing the angle \(\theta\) and then marking off the distance \(r\) along the ray. $$ \left(1, \frac{\pi}{2}\right) $$

Identify the conic and calculate the angle of rotation of axes for the curve described by the equation $$ 3 x^{2}+5 x y-2 y^{2}-125=0 $$

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.\(x=3 \sin t, \quad y=3 \cos t, \quad t=\frac{\pi}{4}\)

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=e^{-2 t}, \quad y=e^{3 t} $$

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.\(x=\cos t, \quad y=8 \sin t, t=\frac{\pi}{2}\)

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=t^{3}, \quad y=3 \ln t $$

For the following exercises, determine the equation of the parabola using the information given. $$ \text { Focus }(4,0) \text { and directrix } x=-4 $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=4 \sec \theta, \quad y=3 \tan \theta $$

For the following exercises, determine the equation of the parabola using the information given. $$ \text { Focus }(0,-3) \text { and directrix } y=3 $$