Give the coordinates of the center, foci, vertices, and asymptotes of the hyperbola with the equation 9x² - 4y² - 90x - 32y = -305.Sketch the graph, and include these points and lines, along with the auxiliary rectangle.

Answer 1

First let's complete the square on
9x² - 4y² - 90x - 32y = -305
9(x^2 - 10x + ....) - 4(y^2 + 16t + ....) = -305
9(x^2 - 10x + 25) - 4(y^2 + 16t + 16) = -305 + 9(25) - 4(16)
9(x - 5)^2 - 4(y + 4)^2 = -144
divide by 144
(x-5)^2 / 16 - (y+4)^2 / 36 = -1

centre is (5,-4)
a = 4
b = 6
c^2 = a^2 + b^2
c = 2√13
hyperbola has vertical major axis

I assume that you can build your auxiliary rectangle around (5,-4),
drawing in your asymptotes (the diagonals), and sketching the curve.

Answer 2

graph