SOLUTION: The hyperbola defined by the equation 3x^2 − 2y^2 + 6x − 8y = 6. And solution. Transverse Axis parallel to: x-axis or y-axis? Graph: Center of the Hyperbola: Vert

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: The hyperbola defined by the equation 3x^2 − 2y^2 + 6x − 8y = 6. And solution. Transverse Axis parallel to: x-axis or y-axis? Graph: Center of the Hyperbola: Vert      Log On


   

Question 1172211: The hyperbola defined by the equation 3x^2 − 2y^2 + 6x − 8y = 6.
And solution.
Transverse Axis parallel to:
x-axis or y-axis?
Graph:
Center of the Hyperbola:
Vertices of the Hyperbola:
Foci of the Hyperbola:
Equations of the Asymptotes:
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


Case 1: If the transverse axis is parallel to the x-axis (branches open right and left), then the equation is

%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2+=+1

Case 2: If the transverse axis is parallel to the y-axis (branches open up and down), then the equation is

%28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2+=+1

In either equation, (h,k) is the center of the hyperbola; 2a is the length of the transverse axis, and 2b is the length of the conjugate axis.

c is the distance from the center to either focus; a, b, and c are related by

c%5E2+=+a%5E2%2Bb%5E2

You need to complete the square in both x and y to put the given equation into one of those forms.

3x%5E2-2y%5E2%2B6x-8y=6
%283x%5E2%2B6x%29-%282y%5E2%2B8y%29=6
3%28x%5E2%2B2x%29-2%28y%5E2%2B4y%29=6
3%28x%5E2%2B2x%2B1%29-2%28y%5E2%2B4y%2B4%29=6%2B3-8
3%28x%2B1%29%5E2-2%28y%2B2%29%5E2=1
%28x%2B1%29%5E2%2F%281%2F3%29-%28y%2B2%29%5E2%2F%281%2F2%29=1

That is in the form of case 1: (h,k) = (-1,-2); a^2=1/3; b^2=1/2.

Transverse Axis parallel to: x-axis or y-axis? See the definition of case 1.

Graph: (I leave that to you)

Center of the Hyperbola: (h,k)

Vertices of the Hyperbola: a units to the right and left of the center

Foci of the Hyperbola: c units to the right and left of the center

Equations of the Asymptotes: the slopes are b/a and -b/a; (h,k) is a point on both asymptotes.