Exam 14: Parametric Equations and Conic Sections

What is the explicit formula for the curve $x=e^{0.1 t}, y=e^{t}, 0 \leq t \leq 1$ ?

Answer the following for the hyperbola $16 y^{2}-128 y-25 x^{2}-50 x=169$ : a) What is the center? b) Does the hyperbola open left-right or up-down? c) What are the values of $a$ and $b$ ?

Write $6.5 e^{x}+1.5 e^{-x}$ as $a \cosh x+b \sinh x$ for some $a$ and $b$ .

What curve is parameterized by the functions $x=4+\sin t$ and $y=4+\cos t$ ? Give an implicit or explicit equation for the curve.

Identify the center and radius of $4 x^{2}-8 x+4 y^{2}-24 y=-35$

Write $7 \cosh (2)+2 \sinh (2)$ in terms of $e^{2}$ and $e^{-2}$ .

Rewrite the equation $15 x^{2}-4 y^{2}=-251+150 x-32 y$ in the form $\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1$ or $\frac{(y-k)^{2}}{b^{2}}-\frac{(x-h)^{2}}{a^{2}}=1$

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 3 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. How many meters\sec. is the mouse traveling while it is on the blade? Round to 2 decimal places.

What is the focus of the parabola $4 y=x^{2}-10 x+25$ ?

Graph $\frac{(y-2)^{2}}{9}-\frac{x^{2}}{4}=1$

In the hyperbola given by the equation $\frac{(x-6)^{2}}{49}-\frac{(y-4)^{2}}{25}=1$ , the left vertex is at (-----------,-------------)

In the ellipse given by the equations $x=7+5 \cos t, y=4+3 \sin t, 0 \leq t \leq 2 \pi$ , the $x$ -values range from a minimum of ----------- to a maximum of ----------- about the midline x=----------.

If the ellipse $x^{2}+9 y^{2}-4 x-72 y+139=0$ is written in standard form $\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$ , then h=-----------,k=----------,a=-----------,and b=---------.

A circle centered at the origin also includes the point $(3,5)$ . What is the radius of the circle? Round to 2 decimal places.

Is the function $4 x^{2}+3 x^{3}=2$ implicit or explicit?

Write a parameterization for one branch of the following hyperbola using hyperbolic functions: $\frac{(y-4)^{2}}{9}-\frac{(x+1)^{2}}{4}=1, y<4$

Find the upper intersection point of the ellipse $\frac{(x+3)^{2}}{9}+\frac{(y-3)^{2}}{16}=1$ and the line $x=$ -2 . Round the $y$ -coordinate to 2 decimal points.

Write an implicit equation for the hyperbola that opens left\right whose asymptotes are the diagonal lines through the corners of a rectangle of width 12 and height 8 centered at the point $(-3,-5)$ .

Write a parameterization for one branch of the following hyperbola using hyperbolic functions: $15 x^{2}-4 y^{2}=-416+180 x-32 y, x<6$

Which of the following are focal points of the hyperbola $\frac{x^{2}}{49}-\frac{y^{2}}{16}=1$ ?