The equation y² - 4y - 4x² + 13 = 0 will produce a hyperbola. How can we tell by simply observing the equation? In what directions do the branches of this hyperbola open? How do you know? Explain. Sketch a graph of this hyperbola, clearly indicating how you have determined the key characteristics (center, vertices, eccentricity, foci). Give the domain and range of this hyperbola.

Find the vertices and locate the foci of the hyperbola with the given equation. Then graph the equation. x² y² 종목이 = 1 4 The vertices of the hyperbola are (Type an ordered pair. Simplify your answer. Use a comma to separate answers as needed.)

Use the information provided to write the standard form equation of each hyperbola.

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 120. The comet follows a parabolic path with directrix x = 85 and vertex at (75, 0). Part A: Write the equation of the planet's orbit in standard form. Show your work. 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.