Explain which conic section this equation and explain how to solve it:

12x^2-18y^2-18x-12y+12=0

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The given equation is 12x^2 - 18y^2 - 18x - 12y + 12 = 0.

To identify the conic section represented by this equation, we need to examine the coefficients of x^2, y^2, xy, x, and y.

The general equation of a conic section is of the form Ax^2 + By^2 + Cxy + Dx + Ey + F = 0, where A, B, C, D, E, and F are coefficients.

In this equation:
- Coefficient of x^2: 12
- Coefficient of y^2: -18
- Coefficient of xy (C): 0 (since there is no xy term)
- Coefficient of x: -18
- Coefficient of y: -12
- Constant term: 12

By comparing the coefficients with the general equation, we can see that A ≠ B (12 ≠ -18), and C = 0. This implies that the equation represents a conic section known as a hyperbola.

To solve for this hyperbola, we can follow these steps:

Step 1: Rewrite the equation
12x^2 - 18y^2 - 18x - 12y + 12 = 0
Divide the equation by 12 to simplify it:
x^2 - (3/2)y^2 - (3/2)x - (1/2)y + 1 = 0

Step 2: Reorder the terms
x^2 - (3/2)x - (3/2)y^2 - (1/2)y + 1 = 0

Step 3: Complete the square for x terms
x^2 - (3/2)x = (3/2)y^2 + (1/2)y - 1
To complete the square, add the square of half the coefficient of x to both sides of the equation:
x^2 - (3/2)x + (3/4)^2 = (3/2)y^2 + (1/2)y - 1 + (3/4)^2
Simplifying further:
x^2 - (3/2)x + 9/16 = (3/2)y^2 + (1/2)y - 16/16

Step 4: Complete the square for y terms
(3/2)y^2 + (1/2)y = x^2 - (3/2)x + 7/16
To complete the square, add the square of half the coefficient of y to both sides of the equation:
(3/2)y^2 + (1/2)y + (1/16)^2 = x^2 - (3/2)x + 7/16 + (1/16)^2
Simplifying further:
(3/2)y^2 + (1/2)y + 1/256 = x^2 - (3/2)x + 7/16 + 1/256

Step 5: Simplify the equations
(3/2)y^2 + (1/2)y + 1/256 = x^2 - (3/2)x + 7/16 + 1/256
Combining the constants:
(3/2)y^2 + (1/2)y + 1/256 = x^2 - (3/2)x + 449/256

Step 6: Rewrite the equations in standard form
x^2 - (3/2)x + 449/256 = (3/2)y^2 + (1/2)y + 1/256

Step 7: Factor out the leading coefficients
x^2 - (3/2)x + 449/256 = (3/2)(y^2 + (1/3)y + 1/192)

Step 8: Simplify the right side
x^2 - (3/2)x + 449/256 = (3/2)(y - (-1/6))^2

Step 9: Rewrite the equation
x^2 - (3/2)x + 449/256 = (3/2)(y + 1/6)^2

Step 10: Define the key properties
The given equation represents a hyperbola with a horizontal transverse axis, as the x term has a positive coefficient and the y term has a negative coefficient.

The center of the hyperbola is at (h, k), where h = 3/4 and k = -1/6.

The squared values in the equation represent the lengths of the transverse and conjugate axes:
- The transverse axis is along the x-axis and has a length of 2(sqrt((3/2) / (3/2))) = 2.
- The conjugate axis is along the y-axis and has a length of 2(sqrt((1/2) / (3/2))) = sqrt(2/3).

Thus, the hyperbola is centered at (3/4, -1/6), has a horizontal transverse axis with a length of 2, and a vertical conjugate axis with a length of sqrt(2/3).

The given equation is a quadratic equation in two variables, x and y. Let's try to identify the conic section by analyzing the equation.

The general equation of a conic section is of the form Ax^2 + By^2 + Cx + Dy + E = 0, where A, B, C, D, and E are constants.

In the given equation, we have:
12x^2 - 18y^2 - 18x - 12y + 12 = 0

Comparing this equation with the general equation, we can see that:
A = 12
B = -18
C = -18
D = -12
E = 12

To identify the conic section, we can calculate the discriminant, which is given by the formula: D = B^2 - 4AC.

Substituting the values, we have:
D = (-18)^2 - 4 * 12 *(-18) = 324 + 864 = 1188

If the discriminant D is positive, the equation represents a hyperbola.
If the discriminant D is zero, the equation represents a parabola.
If the discriminant D is negative, the equation represents an ellipse.

In this case, D = 1188, which is positive. Therefore, the given equation represents a hyperbola.

To solve the equation, we can manipulate the equation to obtain the standard form of a hyperbola. The standard form of a hyperbola with the center (h, k) is given by:

((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1

or

((y - k)^2 / b^2) - ((x - h)^2 / a^2) = 1

where a and b are positive values representing the distances from the center to the vertices in the x and y directions, respectively.

To obtain the standard form, we need to complete the square for both x and y terms. Let's start by completing the square for the x terms:

12x^2 - 18y^2 - 18x - 12y + 12 = 0

Rearranging the terms:
12x^2 - 18x - 18y^2 - 12y + 12 = 0

Grouping the x terms and completing the square:
(12x^2 - 18x) - 18y^2 - 12y + 12 = 0
12(x^2 - (3/2)x) - 18y^2 - 12y + 12 = 0
12(x^2 - (3/2)x + (9/16)) - 18y^2 - 12y + 12 - 12(9/16) = 0

Simplifying:
12(x - 3/4)^2 - 18y^2 - 12y + 12 - 27/4 = 0
12(x - 3/4)^2 - 18y^2 - 12y + 9/4 = 0

Now let's complete the square for the y terms:
12(x - 3/4)^2 - 18y^2 - 12y + 9/4 = 0
12(x - 3/4)^2 - 18(y^2 + (2/3)y) + 9/4 = 0
12(x - 3/4)^2 - 18(y^2 + (2/3)y + 1/9) + 9/4 - 18(1/9) = 0

Simplifying:
12(x - 3/4)^2 - 18(y + 1/3)^2 + 1/4 = 0

Now we have the equation in standard form:
12(x - 3/4)^2 - 18(y + 1/3)^2 + 1/4 = 0

By comparing this with the standard form of a hyperbola, we can determine its center, vertices, and other properties.

The center of the hyperbola is (h, k), which in this case is (3/4, -1/3).

By comparing the coefficients of the squared terms in the standard form equation, we can determine the values of a and b.

a^2 = 1/12
b^2 = 1/18

Taking the square root of a^2 and b^2, we get:
a = sqrt(1/12) = 1 / (2*sqrt(3))
b = sqrt(1/18) = 1 / (3*sqrt(2))

So, the hyperbola has vertices at a distance of a units from the center in the x-direction and b units from the center in the y-direction.

In summary, the given equation 12x^2 - 18y^2 - 18x - 12y + 12 = 0 represents a hyperbola. To solve the equation, we manipulated it to obtain the standard form of a hyperbola and then identified the center, vertices, and other properties of the hyperbola.