Question

In a solar system, two comets pass near the sun, which is located at the origin. Comet E is modeled by quantity y plus 16 end quantity squared over 400 plus x squared over 144 equals 1 and Comet H is modeled by quantity y plus 13 end quantity squared over 144 minus x squared over 25 equals 1 comma where all measurements are in astronomical units.

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 points)

Answer 1

Explanation:

Part B: To determine which comet travels closer to the sun, we need to compare the distances of each comet from the origin (sun). The distance between a point (x, y) and the origin (0, 0) is given by the distance formula: d = sqrt(x^2 + y^2).

For Comet E: The equation for Comet E is (y + 16)^2/400 + x^2/144 = 1. By comparing this equation to the standard form equation of an ellipse, we can determine that the major axis is along the y-axis, with a length of 20 (2a = 20), and the minor axis is along the x-axis, with a length of 12 (2b = 12). Therefore, the distance from the origin to the farthest point on the ellipse (vertex) is a = 10 units. Hence, the farthest distance Comet E can be from the sun is 10 units.

For Comet H: The equation for Comet H is (y + 13)^2/144 - x^2/25 = 1. By comparing this equation to the standard form equation of a hyperbola, we can determine that the major axis is along the y-axis, with a length of 24 (2a = 24), and the minor axis is along the x-axis, with a length of 10 (2b = 10). Therefore, the distance from the origin to the farthest point on the hyperbola (vertex) is a = 12 units. Hence, the farthest distance Comet H can be from the sun is 12 units.

Comparing the distances, we can see that Comet E (10 units) travels closer to the sun compared to Comet H (12 units). Therefore, Comet E travels closer to the sun based on the given equations.

Part C: The sun represents a focus point for both comets. In an ellipse or a hyperbola, the foci are the points inside the shape that help define its shape and position. The foci are related to the distance between the center and the vertices of the shape.

For Comet E: The foci of an ellipse are located along the major axis. Since the major axis is along the y-axis, the foci will be on the y-axis. The distance between the center and the foci is given by c = sqrt(a^2 - b^2). In this case, c = sqrt(400 - 144) = sqrt(256) = 16 units. Therefore, the foci of Comet E will be located 16 units above and below the center (the sun).

For Comet H: The foci of a hyperbola are also located along the major axis. Since the major axis is along the y-axis, the foci will be on the y-axis. The distance between the center and the foci is given by c = sqrt(a^2 + b^2). In this case, c = sqrt(144 + 25) = sqrt(169) = 13 units. Therefore, the foci of Comet H will be located 13 units above and below the center (the sun).

In both comets, the sun (origin) acts as a focus point around which the shape of the comet is determined. For Comet E, the foci are located 16 units above and below the sun, while for Comet H, the foci are located 13 units above and below the sun.

Therefore, the sun represents a focus point for both comets, providing evidence from the equations of the shapes (ellipse and hyperbola) and the distances between the center and the foci.