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 .
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 .
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:
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The points where the planet's orbit intersects the path of the comet are , , and , respectively.