SOLUTION: Convert the equation to standard form. Locate the foci and find the equation of the asymptotes. 16x2 + 128x - 9y2 + 180y

SOLUTION: Convert the equation to standard form. Locate the foci and find the equation of the asymptotes. 16x2 + 128x - 9y2 + 180y - 788 = 0

Question 1121384: Convert the equation to standard form. Locate the foci and find the equation of the asymptotes.
16x2 + 128x - 9y2 + 180y - 788 = 0
Found 2 solutions by Alan3354, greenestamps:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Use ^ for exponents.
Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

(1) Move the constant to the other side of the equation; factor out the leading coefficients in x and y:

(2) Complete the square in both x and y, remembering to add the same amounts to both sides of the equation:

(3) Write the trinomials as binomials squared; divide through by 144 to get "1" on the right side:

or

This is standard form,
.

The center is (h,k) = (-4,10); a=3 is the distance from the center to each vertex; b=4 is the distance from the center to each co-vertex.

The distance from the center to each focus is c, where for a hyperbola . So c=5.

Answers:
(a) The equation is .
(b) The branches of the hyperbola open left and right; The foci are a distance c=5 right or left of the center, at (-9,10) and (1,10).
(c) The slopes of the asymptotes are b/a and -b/a; in this example, 4/3 and -4/3. Use the point-slope form of the equation of a line using each of those slopes, using the center (-4,10) as the point.

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