Answer to Question #255569 in Analytic Geometry for Joshua

Question

According to Kepler’s Laws, planets have elliptical orbits, with the sun

at one of the foci. The farthest Pluto gets from the sun is 7.4 billon

kilometers. The closest it gets to the sun

Answer 1

Let us center the ellipse at $(0,0)$ with the major axis along the $x$- axis.

Using the perihelion, $a - c = 4.4$ billion km.

Using the aphelion, $a + c = 7.4$ billion km.

The semimajor axis is $a = (7.4 + 4.4)/2 = 5.9$ billion km.

$c=7.4-5.9=1.5$ billion km.

$b^2=a^2-c^2=(5.9)^2-(1.5)^2=34.81-2.25$

$=32.56$ (billion km)².

The equation of Pluto's orbit assuming a center at $(0,0)$ is

$$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$$

$x, y, a,$ and $b$ are in billion km

$$\dfrac{x^2}{34.81}+\dfrac{y^2}{32.56}=1$$