Explain which conic section this equation and explain how to solve it:
12x^2-18y^2-18x-12y+12=0
The equation 12x^2 - 18y^2 - 18x - 12y + 12 = 0 represents a conic section. To identify which conic section it is, we can rearrange the equation and group like terms together.
First, let's regroup the equation:
12x^2 - 18y^2 - 18x - 12y + 12 = 0
Rearranging the terms:
12x^2 - 18x - 18y^2 - 12y + 12 = 0
Next, let's group the terms based on their variable and degree:
(12x^2 - 18x) - (18y^2 + 12y) + 12 = 0
Factoring out common factors:
6x(2x - 3) - 6y(3y + 2) + 12 = 0
Now, we can see that there is a difference of squares pattern present:
6x(2x - 3) - 6y(3y + 2) + 12 = 0
This can be rewritten as:
6x(2x - 3) - 6y(3y + 2) + 12 = 0
Factors of 12 can be ±1, ±2, ±3, ±4, ±6, ±12. Using these factors, we can rewrite the equation as follows:
6(x^2 - (3/2)x) - 6(y^2 + (1/2)y) + 12 = 0
Now, we complete the square for both x and y:
6(x^2 - (3/2)x + (9/16)) - 6(y^2 + (1/2)y + (1/16)) + 12 = 0
Expanding:
6(x^2 - (3/2)x + (9/16)) - 6(y^2 + (1/2)y + (1/16)) + 12 = 6(9/16) - 6(1/16)
Simplifying:
6(x - 3/4)^2 - 6(y + 1/4)^2 + 12 = 9/4 - 3/4
6(x - 3/4)^2 - 6(y + 1/4)^2 + 12 = 6/4
Simplifying further:
6(x - 3/4)^2 - 6(y + 1/4)^2 + 12 = 3/2
Finally, we can rewrite the equation in its standard form:
6(x - 3/4)^2 - 6(y + 1/4)^2 = 3/2 - 12
6(x - 3/4)^2 - 6(y + 1/4)^2 = -21/2
Dividing both sides by -21/2:
(x - 3/4)^2 / (-7/4) - (y + 1/4)^2 / (-7/12) = 1
We can now identify that the equation represents a hyperbola. The equation is in the standard form of a hyperbola, with the x term and y term having opposite signs and their squares divided by positive constants.
To solve for this hyperbola, we need to identify the center, vertices, foci, asymptotes, and draw the graph.