Exam 14: Parametric Equations and Conic Sections

An elliptical whispering gallery has a major axis of 46 feet an a minor axis of 38 feet. Where should two speakers position themselves so as to most easily hear each other?

Select the type of curve formed by the parametric equations $x=2+6 \cos 2 t$ and $y=-1+3 \sin 2 t$ .

The center of the circle $5(x-7)^{2}+4(y+4)^{2}-125=-(y+4)^{2}$ is at $(\ldots, \ldots)$ .

Parameterize the hyperbola $\frac{y^{2}}{4^{2}}-\frac{x^{2}}{5^{2}}=1$

Give a parameterization for the horizontal line through the point $(3,4)$ .

Give a parameterization for the vertical line through the point $(2,-4)$ .

Parameterize the hyperbola $\frac{x^{2}}{3^{2}}-\frac{y^{2}}{2^{2}}=1$ .

The polar equation $r=\frac{5}{1-\frac{1}{2} \cos \theta}$ is an ellipse. What is the center of this ellipse?

If the equation $-4 x^{2}+4 y^{2}+8 x-32 y+44=0$ is written in standard form $\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1$ or $\frac{(y-k)^{2}}{b^{2}}-\frac{(x-h)^{2}}{a^{2}}=1$ , then h =-----------,k=----------,a=---------,and b =--------------.

What is the focus of the parabola $y=0.5 x^{2}$ ?

The line between $(0,6)$ and $(-4,11)$ can be parameterized by $x=-4 t, y=6+5 t$ , $0 \leq t \leq 1$ .

The line between $(0,4)$ and $(-4,11)$ can be parametrized by $x=-4 t, y=4+11 t$ , $0 \leq t \leq 1$ .

The ellipse given by the equation $\frac{(x-7)^{2}}{36}+\frac{(y-1)^{2}}{25}=1$ has a minor axis of length -----------.

Let an ellipse be parameterized by the equations x=4+2t y=3+4t a) What is the center? b) What is the length of the major axis? c) What is the length of the minor axis? d) Find an implicit equation of the curve.

A hyperbolic mirror has equation $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ . You wish to shoot a beam of light from the point $(-36,21)$ to reflect off the mirror and arrive at the focus closest to the point $(-36,21)$ . At what slope should you aim the beam of light? Round to 2 decimal places.

If the circle $x^{2}+y^{2}+12 x-16 y+51=0$ is put into standard form $(x-h)^{2}+(y-k)^{2}=r^{2}$ , then $h=\ldots, k=\ldots$ , and $r=$ ------.

The center of the circle parameterized by $x=4+9 \cos t, y=-1+9 \sin t, 0 \leq t \leq 2 \pi$ is at (----------,---------).

Write $4\left(e^{11 x}+e^{3 x}-e^{-3 x}-e^{-11 x}\right)$ as $a \cosh (b x) \sinh (c x)$ for some $a, b$ , and $c$ .

What is the center of the hyperbola $-4 x^{2}+9 y^{2}+8 x-54 y+41=0$ ?

Write a parameterization of one branch of the following hyperbola using hyperbolic functions : $25 x^{2}-9 y^{2}-90 y=350$