Explain which conic section this equation and explain how to solve it:
12x^2-18y^2-18x-12y+12=0
When the x^2 and y^2 terms have opposite signs, you have a hyperbola. For the location and the asymptotes, try completing the squares of the x and y polynomials.
The given equation, 12x^2 - 18y^2 - 18x - 12y + 12 = 0, represents a conic section. To determine which type of conic section it is, we need to simplify the equation and analyze its key components.
Step 1: Rearrange the equation
Start by rearranging the equation to group the x-terms and y-terms separately:
12x^2 - 18y^2 - 18x - 12y + 12 = 0
(12x^2 - 18x) - (18y^2 + 12y) + 12 = 0
Step 2: Factor the quadratic terms
Factor out the common factors for the x-terms and y-terms:
6x(2x - 3) - 6y(3y + 2) + 12 = 0
Step 3: Complete the square for x and y
We can complete the square for the x-terms by adding and subtracting a constant value. Similarly, we can complete the square for the y-terms as well:
6(2x - 3) - 6(3y + 2) + 12 = 0
12x - 18 - 18y - 12 + 12 = 0
12x - 18y - 18 = 0
Step 4: Analyze the equation
The simplified equation is 12x - 18y - 18 = 0. This equation matches the standard form of a linear equation (Ax + By + C = 0), which means that the given equation represents a straight line.
Therefore, the answer is that the given equation does not represent a conic section; it represents a straight line.
To solve the equation, you can find various points on the line by substituting values for either x or y. Since this is a linear equation, there are infinitely many solutions. To find specific points on the line, you can choose specific values for x (or y) and solve for the corresponding y (or x) values. For example, if you choose x = 0, you can solve for y:
12(0) - 18y - 18 = 0
-18y - 18 = 0
-18y = 18
y = -1
Thus, when x = 0, y = -1 is a point on the line represented by the given equation. Similarly, you can choose different x values and find the corresponding y values to obtain more points on the line.