a) The equation of the planet's orbit is
.
b) The equation for the path of the comet is
.
c) The points where the planet's orbit intersects the path of the comet are
,
,
and
, respectively.
<h3>
Application of the equations of the circle and the parabola in orbit description</h3>
a) The equation of the circle in <em>standard</em> form is described below:
(1)
Where:
- Coordinates of the center.
- Radius of the orbit.
Please notice that the diameter is two times the radius of the orbit. Now we derive the expression for the orbit of the planet: (
,
,
)
(2)
The equation of the planet's orbit is
. ![\blacksquare](https://tex.z-dn.net/?f=%5Cblacksquare)
b) According to the statement, the parabola has the x-axis as its axis of symmetry and grows in the -x direction.
(3)
Where:
- Coordinates of the vertex.
- Distance from the directrix to the vertex.
Now we derive the expression for the path of the comet: (
,
,
)
(4)
The equation for the path of the comet is
. ![\blacksquare](https://tex.z-dn.net/?f=%5Cblacksquare)
c) An efficient approach consist in plotting each expression in a <em>graphic</em> tool, whose outcome is presented in the image attached below. There are four points where the planet's orbit intersects the path of the comet:
-
![(x_{1}, y_{1}) = (-49.945, 2.345)](https://tex.z-dn.net/?f=%28x_%7B1%7D%2C%20y_%7B1%7D%29%20%3D%20%28-49.945%2C%202.345%29)
-
![(x_{2}, y_{2}) = (-49.945, -2.345)](https://tex.z-dn.net/?f=%28x_%7B2%7D%2C%20y_%7B2%7D%29%20%3D%20%28-49.945%2C%20-2.345%29)
-
![(x_{3}, y_{3}) = (49.995, 0.707)](https://tex.z-dn.net/?f=%28x_%7B3%7D%2C%20y_%7B3%7D%29%20%3D%20%2849.995%2C%200.707%29)
-
![(x_{1}, y_{2}) = (49.995, -0.707)](https://tex.z-dn.net/?f=%28x_%7B1%7D%2C%20y_%7B2%7D%29%20%3D%20%2849.995%2C%20-0.707%29)
The points where the planet's orbit intersects the path of the comet are
,
,
and
, respectively. ![\blacksquare](https://tex.z-dn.net/?f=%5Cblacksquare)