Exam 14: Parametric Equations and Conic Sections

Describe the graph of the parametric equations $x=e^{6 t}, y=e^{12 t},-\infty<t<\infty$ .

Find the equation of a parabola centered at the origin with focal point $(0,3)$ .

Write the ellipse $4 x^{2}+16 x+8 y^{2}-64 y=-139$ in the form $\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$ .

$(x-3)^{2}+(y+5)^{2}=9$ is a circle of radius 3 centered at $(3,-5)$ .

In the hyperbola given by the equation $\frac{(x-4)^{2}}{25}-\frac{(y-5)^{2}}{4}=1$ , the asymptote that slopes upward has equation $y=\ldots x+\ldots$ . Round both answers to 2 decimal places.

In the ellipse given by the equation $\frac{(x-7)^{2}}{16}+\frac{(y-7)^{2}}{25}=1$ , the $x$ -values range from a minimum of --------- to a maximum of ------------about the midline x=------------.

The ellipse given by the equations $x=2+3 \cos t, y=2+5 \sin t, 0 \leq t \leq 2 \pi$ , has a height of ---------- units and a width of ---------- units.

Give a parameterization for the upper half circle of radius 3 centered at $(7,-2)$ , and starting at $(10,-2)$

What is the point of intersection in the first quadrant of the curves $x^{2}+y^{2}=\frac{1}{2}$ and $x^{2}-y^{2}=\frac{1}{4} ?$ Round each coordinate to 2 decimal places.

What is the explicit formula for the curve $x=t^{5}+3, y=t^{4}+5$ ?

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 7 meters long and the windmill makes one complete revolution every 28 seconds in a counterclockwise direction. The mouse starts in the 3 o'clock position and, 10.5 seconds later, loses its hold and flies off. Assume that when the mouse leaves the blade it moves along a straight line tangent to the circle on which it was previously moving. The equation of that line is y=--------+-----------x. Round to 2 decimal places if necessary.

$P$ is a point on the unit circle, and $Q$ and $R$ lie on the ellipse shown below. The equation of the ellipse is given by $\frac{x^{2}}{4}+y^{2}=1$ . What is the $x$ -coordinate of $Q$ ?

Write a parameterization of one branch of the following hyperbola using hyperbolic functions : $\frac{(x-4)^{2}}{25}-\frac{(y-7)^{2}}{9}=1, x>4$

Find the equation of the ellipse centered at $(1,-3)$ with horizontal axis of length 8 and vertical axis of length 12 .

Let $x=\sqrt{3 t}, y=8 t+8,0 \leq t \leq \infty$ . a) Graph the curve. b) Eliminate the parameter to obtain an equation for $y$ as a function of $x$ .

Find a parameterization of the curve $\frac{(x-2)^{2}}{25}+\frac{(y-3)^{2}}{16}=1$ so that the curve is traversed in a counter-clockwise direction starting at $(7,3)$ .

Choose the parametric equations that describe the curve

a) Parameterize $\frac{x^{2}}{25}-y^{2}=1$ . b) Which $t$ values give the left half of the hyperbola?

What are the endpoints for the curve $x=e^{0.05 t}, y=e^{t}, 0 \leq t \leq 1$ ? Mark both correct answers.

Write $11 \cosh x+3 \sinh x$ in terms of $e^{x}$ and $e^{-x}$ .