2x^2 - 12x - 3y^2 - 24y +60 = 0
What are we doing with this hyperbola ?
To solve the equation 2x^2 - 12x - 3y^2 - 24y + 60 = 0, we need to use a method called factoring or completing the square.
Let's begin by grouping the terms with x together and the terms with y together:
2x^2 - 12x - 3y^2 - 24y + 60 = 0
(2x^2 - 12x) - (3y^2 + 24y) + 60 = 0
Now, let's factor out the common factors within each parentheses:
2(x^2 - 6x) - 3(y^2 + 8y) + 60 = 0
Next, we want to complete the square for both x and y terms separately. To complete the square, we need to take half of the coefficient of the middle term, square it, and add it to both sides of the equation.
For the x terms:
We have (x^2 - 6x). Taking half of -6, we get -3. To complete the square, we add (-3)^2 = 9 to both sides of the equation:
2(x^2 - 6x + 9) - 3(y^2 + 8y) + 60 = 0 + 2(9)
Simplifying this equation further:
2(x - 3)^2 - 3(y^2 + 8y) + 60 = 18
For the y terms:
We have (y^2 + 8y). Taking half of 8, we get 4. To complete the square, we add (4)^2 = 16 to both sides of the equation:
2(x - 3)^2 - 3(y^2 + 8y + 16) + 60 = 18 + 3(16)
Simplifying this further:
2(x - 3)^2 - 3(y + 4)^2 + 60 = 66
Now, we can rewrite the equation in the standard form:
2(x - 3)^2 - 3(y + 4)^2 = 6
Based on this equation, we can determine that the general equation represents an ellipse. The center of the ellipse is at (3, -4) and it has a major axis along the x-axis.
To find more information, we can compare the standard form equation with the general equation of an ellipse:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
From this comparison, we can conclude that: a^2 = 3, b^2 = 2/3, h = 3, k = -4.
Therefore, we have obtained the center, major axis, and values of a and b for the given ellipse equation.