이차곡선

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 $(3,6)$

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.