The equation of the circular orbit and the parabolic path of the comet are
given by the center, diameter, vertex, and directrix.
Part A: The equation of the planet's orbit is; <u>x² + y² = 50²</u>
Part B: The equation of the comet's orbit is;
Part C: <u>The planet's orbit and the comet's path do not intersect</u>.
Reasons:
Part A: The center of the orbit = The origin (0, 0)
The diameter of the orbit = 100
Therefore;
The radius of the planet's orbit, r = 50
The equation of a circle with center (h, k) is; (x - h)² + (y - k)² = r²
Therefore, the equation of the circle is (x - 0)² + (y - 0)² = r²
Which gives;
<u>x² + y² = </u><u>50²</u>
Part B: The directrix of the parabolic path is x = 70
The vertex = (60, 0)
Therefore, -p = 70
Which gives;
The equation of the comet's path in standard form is
Part C: At the point where the path intersect, we have;
Therefore;
Let y² = a, we get;
a ≈ -2015.693 and a ≈ -42784.31
y ≈ √(-2015.693) or y ≈ √-42784.31) (imaginary numbers)
<u>The planet's orbit does not intersect the path of the comet</u>.
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