Explain which conic section this equation and explain how to solve it:
12x^2-18y^2-18x-12y+12=0
let's complete the square to get it into standard form
12(x^2 - (3/2)x + .....) - 18(y^2 + (2/3)y + ...) = -12
12(x^2 - (3/2)x + 9/16) - 18(y^2 + (2/3)y + 1/9) = -12 + 9/16 + 1/9
12(x - 3/4)^2 - 18(y + 1/3)^2 = -1631/144
divide by 1631/144
(x-3/4)^2/(1728/1631) - (y+1/3)^2/(2592/1631) = -1
I am sure you can determine the properties of the vertical hyperbola from there.
Check my arithmetic.
The given equation, 12x^2 - 18y^2 - 18x - 12y + 12 = 0, represents a conic section. To determine which type of conic section it is and solve the equation, we can follow these steps:
Step 1: Rearrange the equation
Begin by rearranging the terms of the equation to simplify it and isolate the variables on one side:
12x^2 - 18y^2 - 18x - 12y + 12 = 0
Rearranging the terms gives us:
12x^2 - 18y^2 - 18x - 12y = -12
Step 2: Group the variables and complete the square
Group together the x-terms and the y-terms:
12x^2 - 18x - 18y^2 - 12y = -12
Now, focus on completing the square for the x-terms and y-terms separately:
For the x-terms:
To complete the square for the x-terms, divide the coefficient of x by 2, square it, and add it to both sides of the equation. In this case, the coefficient of x is -18:
12x^2 - 18x = -12 + (9)^2
This simplifies to:
12(x^2 - 3x) = -12 + 81
For the y-terms:
Perform the same steps as for the x-terms. In this case, the coefficient of y is -12:
-18y^2 - 12y = -12 + (6)^2
This simplifies to:
-18(y^2 + 2y) = -12 + 36
Step 3: Factor perfect squares
Now, factor the perfect squares obtained in the previous step:
For the x-term:
12(x^2 - 3x) = 69
We can rewrite x^2 - 3x as (x - 3/2)^2 - (3/2)^2, using the formula (a - b)^2 = a^2 - 2ab + b^2. Therefore, the equation becomes:
12((x - 3/2)^2 - (3/2)^2) - 18(y^2 + 2y) = 24
For the y-term:
-18(y^2 + 2y) = 24
We can rewrite y^2 + 2y as (y + 1)^2 - 1^2, using the same perfect square formula. Therefore, the equation becomes:
12((x - 3/2)^2 - (3/2)^2) - 18((y + 1)^2 - 1^2) = 24
Step 4: Simplify and standardize the equation
Now we can simplify and standardize the equation further:
12(x - 3/2)^2 - 9 - 18(y + 1)^2 + 18 = 24
12(x - 3/2)^2 - 18(y + 1)^2 = 15
Dividing both sides of the equation by 15 gives us:
(x - 3/2)^2 / (15/12) - (y + 1)^2 / (15/18) = 1
Simplifying the fractions:
(x - 3/2)^2 / (5/4) - (y + 1)^2 / (5/6) = 1
We can see that the equation is in the standard form for a hyperbola. The center of the hyperbola is at the point (3/2, -1), and the x-axis and y-axis are the transverse and conjugate axes, respectively. The square root of the denominators (5/4 and 5/6) determines the length of the transverse and conjugate axes, respectively.