1.5 Conic sections  (Page 11/23)

A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet across, find the depth.

Whispering galleries are rooms designed with elliptical ceilings. A person standing at one focus can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet and the foci are located 30 feet from the center, find the height of the ceiling at the center.

A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus and the other focus is 80 feet away, what is the length and the height at the center of the gallery?

Halley’s Comet: length of major axis = 35.88, eccentricity = 0.967

Hale-Bopp Comet: length of major axis = 525.91, eccentricity = 0.995

Mars: length of major axis = 3.049, eccentricity = 0.0934

Jupiter: length of major axis = 10.408, eccentricity = 0.0484

The rectangular coordinates of the point $\left(4,\frac{5\pi }{6}\right)$ are $\left(2\sqrt{3},\mathrm{-2}\right).$

The equations $x=\text{cosh}(3t),$ $y=2\text{sinh}(3t)$ represent a hyperbola.

The arc length of the spiral given by $r=\frac{\theta }{2}$ for $0\le \theta \le 3\pi$ is $\frac{9}{4}{\pi }^3.$

Given $x=f(t)$ and $y=g(t),$ if $\frac{dx}{dy}=\frac{dy}{dx},$ then $f(t)=g(t)+\text{C,}$ where C is a constant.

$x=1+t,$ $y=t^2-1,$ $\mathrm{-1}\le t\le 1$

$x=e^t,$ $y=1-e^{3t},$ $0\le t\le 1$

$x=\text{sin}\theta ,$ $y=1-\text{csc}\theta ,$ $0\le \theta \le 2\pi$

$x=4\text{cos}\varphi ,$ $y=1-\text{sin}\varphi ,$ $0\le \varphi \le 2\pi$

$r=4\text{sin}\left(\frac{\theta }{3}\right)$

$r=5\text{cos}\left(5\theta \right)$

$x+y=5$

$y^2=4+x^2$

$x=\text{ln}(t),$ $y=t^2-1,$ $t=1$

$r=3+\text{cos}\left(2\theta \right),$ $\theta =\frac{3\pi }{4}$

Find $\frac{dy}{dx},$ $\frac{dx}{dy},$ and $\frac{d^{2}x}{d{y}^2}$ of $y=\left(2+e^{\text{−}t}\right),$ $x=1-\text{sin}(t)$

$x=t^2,$ $y=\text{ln}(t),$ $0\le t\le e$

$r=1-\text{sin}\theta$ in the first quadrant

$x=3t+4,$ $y=9t-2,$ $0\le t\le 3$

$r=6\text{cos}\theta ,$ $0\le \theta \le 2\pi .$ Check your answer by geometry.

A parabola with focus $(2,\mathrm{-5})$ and directrix $x=6$

An ellipse with a major axis length of 10 and foci at $\left(\mathrm{-7},2\right)$ and $\left(1,2\right)$

A hyperbola with vertices at $(3,\mathrm{-2})$ and $(\mathrm{-5},\mathrm{-2})$ and foci at $(\mathrm{-2},\mathrm{-6})$ and $(\mathrm{-2},4)$

$r=\frac{6}{1+3\text{cos}(\theta )}$

$r=\frac{4}{3-2\text{cos}\theta }$

$r=\frac{7}{5-5\text{cos}\theta }$

Determine the Cartesian equation describing the orbit of Pluto, the most eccentric orbit around the Sun. The length of the major axis is 39.26 AU and minor axis is 38.07 AU. What is the eccentricity?

The C/1980 E1 comet was observed in 1980. Given an eccentricity of 1.057 and a perihelion (point of closest approach to the Sun) of 3.364 AU, find the Cartesian equations describing the comet’s trajectory. Are we guaranteed to see this comet again? ( Hint : Consider the Sun at point $(0,0).)$