Lyra is designing a model of a solar system with a planet and a comet. The planet has a circular orbit, centered at the origin Lyra is designing a model of a solar system with a planet and a comet. The planet has a circular orbit, centered at the origin with a diameter of 100. The comet follows a diameter of 100. The comet follows

a parabolic path with directrix x = 70 and vertex at (60,0).

Part A: Write the equation of the planet's orbit in standard form. Show your work (2 points)

Part B: Write the equation of the comet's path in standard form Show your work.

Part C: Identify all points where the planet's orbit intersects the path of the comet. Show your work and round answers to the hundredths place.

Question

19-01-2022
Mathematics

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Lyra is designing a model of a solar system with a planet and a comet. The planet has a circular orbit, centered at the origin Lyra is designing a model of a solar system with a planet and a comet. The planet has a circular orbit, centered at the origin with a diameter of 100. The comet follows a diameter of 100. The comet follows

a parabolic path with directrix x = 70 and vertex at (60,0).

Part A: Write the equation of the planet's orbit in standard form. Show your work (2 points)

Part B: Write the equation of the comet's path in standard form Show your work.

Part C: Identify all points where the planet's orbit intersects the path of the comet. Show your work and round answers to the hundredths place.

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xero099 xero099 · Answer 1 · 2022-02-02T12:13:33+01:00

a) The equation of the planet's orbit is [tex]x^{2} + y^{2} = 2500[/tex].

b) The equation for the path of the comet is [tex]x = -40\cdot y^{2} + 60[/tex].

c) The points where the planet's orbit intersects the path of the comet are [tex](x_{1}, y_{1}) = (-49.945, 2.345)[/tex], [tex](x_{2}, y_{2}) = (-49.945, -2.345)[/tex], [tex](x_{3}, y_{3}) = (49.995, 0.707)[/tex] and [tex](x_{1}, y_{2}) = (49.995, -0.707)[/tex], respectively.

Application of the equations of the circle and the parabola in orbit description

a) The equation of the circle in standard form is described below:

[tex](x-h)^{2}+(y-k)^{2} = {r}^{2}[/tex] (1)

Where:

[tex]h,k[/tex] - Coordinates of the center.

[tex]r[/tex] - 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: ([tex]h = 0[/tex], [tex]k = 0[/tex], [tex]r = 50[/tex])

[tex]x^{2} + y^{2} = 2500[/tex] (2)

The equation of the planet's orbit is [tex]x^{2} + y^{2} = 2500[/tex]. [tex]\blacksquare[/tex]

b) According to the statement, the parabola has the x-axis as its axis of symmetry and grows in the -x direction.

[tex]x - h = 4\cdot c \cdot (y-k)^{2}[/tex] (3)

Where:

[tex]h,k[/tex] - Coordinates of the vertex.
[tex]c[/tex] - Distance from the directrix to the vertex.

Now we derive the expression for the path of the comet: ([tex]h = 60[/tex], [tex]k = 0[/tex], [tex]c = -10[/tex])

[tex]x = -40\cdot y^{2} + 60[/tex] (4)

The equation for the path of the comet is [tex]x = -40\cdot y^{2} + 60[/tex]. [tex]\blacksquare[/tex]

c) An efficient approach consist in plotting each expression in a graphic 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:

[tex](x_{1}, y_{1}) = (-49.945, 2.345)[/tex]
[tex](x_{2}, y_{2}) = (-49.945, -2.345)[/tex]
[tex](x_{3}, y_{3}) = (49.995, 0.707)[/tex]
[tex](x_{1}, y_{2}) = (49.995, -0.707)[/tex]

The points where the planet's orbit intersects the path of the comet are [tex](x_{1}, y_{1}) = (-49.945, 2.345)[/tex], [tex](x_{2}, y_{2}) = (-49.945, -2.345)[/tex], [tex](x_{3}, y_{3}) = (49.995, 0.707)[/tex] and [tex](x_{1}, y_{2}) = (49.995, -0.707)[/tex], respectively. [tex]\blacksquare[/tex]