PLEASE SOLVE COMPLETELY WITH A BRIEF EXPLANATION FOR STEPS

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 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 E: Write the equation of the comet's path in standard form, Show your work, (4 points) 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. (4 points) A system of equations is made up of an ellipse and a hyperbole. Part A: Create the equation of an ellipse centered at the origin, with a vertical major axis of 12 units and a minor axis of 6 units. Show your work. (3 points) Part B: Create the equation ofa hyperbola centered at the origin, with a horizontal transverse axis, vertex at (5, 0), and asymptotes 01 y - tgx. Show your work. Part 0: Determine the domain of each conic section and use it to explain why there is no solution to the system. (3 points) An architect has designed two tunnels. Tunnel A Is modeled by x2 + y2 + 20x7 69 = O, and tunnel B is modeled by'x2 7 40x+ 16y + 160 = 0, where all measurements are in feet. The architect wants to verify whether a truck that is 8 feet wide and 13.5 feet high can pass through the tunnels. Part A: Write the equation for Tunnel A in standard form and determine the conic section. Show your work. (4 points) Part B: Write the equation for Tunnel B in standard form and determine the conic section. Show your work. (4 paints) Part C: Determine the maximum height of each tunnel. Is the truck able to pass through either tunnel without damage? If so, which tunnel(s) and why? Show your work. {7 points) In a solar system. two comets pass near the sun, which l5 located at the origin. Comet E is modeled by $4. 1:: 1 and Comet H is modeled by %_g -1. where all measurements are in astronomical units. Part A: What are the vertices for the path of Comet E? Show your work. (4 points) Part B: Which comet travels closer to the sun? Give evidence to support your answer. (5 points) Part C: What key feature does the sun represent for both comets? Give evidence to support your answer. (6 paints)