SOLUTION: Give the coordinates of the center, foci and vertices with equation 9x2 - 4y2 - 90x - 32y = -305.

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Question 1119419: Give the coordinates of the center, foci and vertices with equation 9x2 - 4y2 - 90x - 32y = -305.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


(1) Factor out the leading coefficients in both x and y.

9%28x%5E2-10x%29-4%28y%5E2%2B8y%29+=+-305

(2) Complete the square in both x and y, keeping the equation balanced.

9%28x%5E2-10x%2B25%29+-+4%28y%5E2%2B8y%2B16%29+=+-305%2B225-64+=+-144

9%28%28x-5%29%5E2%29+-+4%28%28y%2B4%29%5E2%29+=+-144

(3) Divide both sides by the constant on the right.

-%28%28x-5%29%5E2%29%2F16%2B%28%28y%2B4%29%5E2%29%2F36+=+1
%28%28y%2B4%29%5E2%29%2F36-%28%28x-5%29%5E2%29%2F16+=+1
%28%28y%2B4%29%5E2%29%2F%286%5E2%29+-+%28%28x-5%29%5E2%29%2F%284%5E2%29+=+1

This is a hyperbola with the branches opening up and down; the standard form of the equation is

%28%28y-k%29%5E2%29%2F%28a%5E2%29+-+%28%28x-h%29%5E2%29%2F%28b%5E2%29+=+1

(h,k) is the center; a is the distance from the center to each end of the transverse axis (between the two vertices); b is the distance from the center to each end of the conjugate axis.

So in this hyperbola the center is (5,-4), and the distance from the center to each vertex is 6; that makes the two vertices (5,-10) and (5,2).

c is the distance from the center to each focus; for a hyperbola, c%5E2+=+a%5E2%2Bb%5E2. So for this hyperbola, c+=+sqrt%2816%2B36%29+=+sqrt%2852%29+=+2%2Asqrt%2813%29. The two foci are then (5,-4-2*sqrt(13)) and (5,-4+2*sqrt(13)).

Answers:
center (5,-4)
vertices: (5,-10) and (5,2)
foci: (5,-4-2*sqrt(13)) and (5,-4+2*sqrt(13))

A graph....