Exam 14: Parametric Equations and Conic Sections

The parameterization for the lower half of the hyperbola $16(y+8)^{2}-9(x-6)^{2}=144$ using hyperbolic functions is $x=a+b \sinh t, y=c+d \cosh t,-\infty<t<\infty$ , where a =----------,b =--------,c =------,and d =----------.

A mouse hanging on to the end of the windmill blade shown below has coordinates given by $x=A \cos (k t), y=A \sin (k t)$ , where the origin is at the center of the blades, $x$ and $y$ are in meters, and $t$ is in seconds. The blades are 6 meters long and the windmill makes one complete revolution every 20 seconds in a counterclockwise direction. The mouse starts in the 3 o'clock position and, 7.5 seconds later, loses its hold and flies off. What is $A$ ?

In the hyperbola given by the equations $x=4+6 \sec t, y=7+3 \tan t, 0 \leq t \leq 2 \pi$ , the asymptote that slopes upward has equation y=-------------x+----------. Round both answers to 2 decimal places.

Select the equations that describe the graph

Find the center, vertices, and asymptotes for the following hyperbola : $\frac{(x-2)^{2}}{36}-\frac{(y+5)^{2}}{25}=1$

Select the type of curve formed by the parametric equations $x=2-\frac{4}{\cos 5 t}$ and $y=2-6 \sin 5 t$ .

If the circle $x^{2}+y^{2}+14 x-6 y+9=0$ is converted to the parametric equations $x=h+r \cos t, y=k+r \sin t, 0 \leq t \leq 2 \pi$ , then h=------------,k=----------,and r=------------.

Parameterize the circle of diameter 4 centered at $(6,-3)$ traversed counterclockwise starting at $(8,-3)$ .

What is the direction of the circle parameterized by $x=4 \cos t, y=4 \sin t, 0 \leq t \leq 2 \pi$ ?

Parameterize the ellipse $\frac{x^{2}}{4^{2}}+\frac{(y-2)^{2}}{4^{2}}=1$ so that the parameterization starts at the point $(4,2)$ and travels clockwise.

Write $\frac{1}{4}\left(e^{12 x}+e^{2 x}-e^{-2 x}-e^{-12 x}\right)$ as $\cosh (a x) \sin (b x)$ for some $a$ and $b$ .

The ellipse given by the equation $\frac{(x-1)^{2}}{25}+\frac{(y-8)^{2}}{16}=1$ has a height of ---------- units and a width of ---------- units.

Is $x=3$ an asymptote of the hyperbola shown below?

Which of the following are focal points of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{4}=1$ ?

In the hyperbola given by the equations $x=2+7 \sec t, y=6+7 \tan t, 0 \leq t \leq 2 \pi$ , the left vertex is at (-----------,--------------).

Is $(4,-1)$ a vertex of the hyperbola shown below?

Find the center of the ellipse and the lengths of the major and minor axes: $25 x^{2}-10 x+16 y^{2}+96 y=231$

Choose the parametric equations that describe the curve

Parameterize the ellipse centered at the point $(-3,4)$ with horizontal axis of length 4 and vertical axis of length 10 .

Write $2 \sinh ^{2} x-6 \cosh ^{2} x$ in terms of $e$ .