매개변수와 극좌표 전체 문제

In Exercises 35–38, find the equation of the hyperbola defined by the given information. Sketch the hyperbola.

Foci: $(\pm 3,0)$; vertices: $(\pm 2, 0)$

In Exercises 35–38, find the equation of the hyperbola defined by the given information. Sketch the hyperbola.

Foci: $(0,\pm 3)$; vertices: $(0,\pm 2)$

In Exercises 35–38, find the equation of the hyperbola defined by the given information. Sketch the hyperbola.

Foci: $(-2,3)$ and $(8,3)$; vertices: $(-1,3)$ and $(7,3)$

In Exercises 35–38, find the equation of the hyperbola defined by the given information. Sketch the hyperbola.

Foci: $(3,-2)$ and $(3,8)$; vertices: $(3,0)$ and $(3,6)$

In Exercises 39–42, write the equation of the hyperbola in standard form.

$3x^2-4y^2=12$

In Exercises 39–42, write the equation of the hyperbola in standard form.

$3x^2-y^2+2y=10$

In Exercises 39–42, write the equation of the hyperbola in standard form.

$x^2-10y^2+40y=30$

In Exercises 39–42, write the equation of the hyperbola in standard form.

$(4y-x)(4y+x)=4$

Consider the ellipse given by $\displaystyle \frac{(x-1)^2}{4}+\frac{(y-3)^2}{12}=1$.

(a) Verify that the foci are located at $(1,3\pm 2\sqrt{2})$. (b) The points $P_1 = (2,6)$ and $P_2 = (1+\sqrt{2},3+\sqrt{6}) \approx (2.414,5.449)$ lie on the ellipse. Verify that the sum of distances from each point to the foci is the same.

Johannes Kepler discovered that the planets of our solar system have elliptical orbits with the Sun at one focus. The Earth's elliptical orbit is used as a standard unit of distance; the distance from the center of Earth's elliptical orbit to one vertex is 1 Astronomical Unit, or A.U.

The following table gives information about the orbits of three planets.

$$\begin{array}{ccc} & \text{Distance from center to vertex} & \text{eccentricity} \\ \hline \text{Mercury} & \text{0.387 A.U.} & \text{0.2056} \\ \text{Earth} & \text{1 A.U.} & \text{0.0167} \\ \text{Mars} & \text{1.524 A.U.} & \text{0.0934}\end{array}$$

(a) In an ellipse, knowing $c^2=a^2-b^2$ and $e=c/a$ allows us to find $b$ in terms of $a$ and $e$. Show $b=a\sqrt{1-e^2}$. (b) For each planet, find equations of their elliptical orbit of the form $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. (This places the center at $(0,0)$, but the Sun is in a different location for each planet.) (c) Shift the equations so that the Sun lies at the origin. Plot the three elliptical orbits.

T/F: When sketching the graph of parametric equations, the $x$ and $y$ values are found separately, then plotted together.

The direction in which a graph is “moving” is called the _____ of the graph.

An equation written as $y=f(x)$ is written in _____ form.

Create parametric equations $x=f(t)$, $y=g(t)$ and sketch their graph. Explain any interesting features of your graph based on the functions $f$ and $g$.

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=t^2+t$, $y=1-t^2$, $-3\leq t\leq 3$

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=1$, $y=5\sin t$, $-\pi/2\leq t\leq \pi/2$

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=t^2$, $y=2$, $-2\leq t\leq 2$

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=t^3-t+3$, $y=t^2+1$, $-2\leq t\leq 2$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=t^3-2t^2$, $y=t^2$, $-2\leq t \leq 3$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=1/t$, $y=\sin t$, $0< t \leq 10$