Given the following equation for a hyperbola: 16x^2-9y^2-64x-90y-305=0
write the equation in standard form and graph it.
(label the center vertices and slanted asymptotes)
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Step-by-step explanation:

16x² − 9y² − 64x − 90y − 305 = 0

Group the x's together and the y's together.

16x² − 64x − 9y² − 90y = 305

Factor:

16 (x² − 4x) − 9 (y² − 10y) = 305

Complete the squares:

16 (x² − 4x + 4) − 9 (y² − 10y + 25) = 305 + 16(4) − 9(25)

16 (x − 2)² − 9 (y − 5)² = 144

Divide by 144:

(x − 2)²/9 − (y − 5)²/16 = 1

This is a horizontal hyperbola (the x term is positive).

The center is (2, 5).

The vertices are (-1, 5) and (5, 5) (plug in y = 5).

The slant asymptotes are y = ±16/9 (x − 2) + 5.

Graph:

desmos.com/calculator/tkcwnlb1ga

answer: both chords and diameters have two endpoints on a circle. diameters must intersect the center of a circle.