Write the following equation in standard form and identify the type of conic:
9y^2 - 4x^2 + 8x + 18y + 41 = 0
9y^2 - 4x^2 + 8x + 18y + 41 = 0
4x^2 - 8x - 9y^2 - 18y - 41 = 0
4(x^2 - 2x + ......) - 9(y^2 + 2y + .... ) = 41
let's "complete these squares"
4(x^2 - 2x + 1) - 9(y^2 + 2y + 1 ) = 41 + 4 - 9
4(x-1)^2 - 9(y+1)^2 = 36
divide by 36
(x-1)^2/9 - (y+1)^2/4 = 1
If you are studying conics you should now be able to identify the conic and state its properties.
To write the equation in standard form and identify the type of conic, we need to rearrange the equation and complete the square for both the x and y terms.
First, let's group the x and y terms together:
9y^2 + 18y - 4x^2 + 8x + 41 = 0
Now, let's complete the square for the x terms:
9y^2 + 18y - 4x^2 + 8x = -41
Next, we need to move the constant term to the right side of the equation:
9y^2 + 18y - 4x^2 + 8x + 41 = 0
Completing the square for the x-terms involves two steps:
1. Divide the coefficient of x (which is 8) by 2, and then square the result: (8/2)^2 = 16.
2. Add the result obtained in step 1 to both sides of the equation.
9y^2 + 18y - 4x^2 + 8x + 16 = 0
Now, let's rewrite the expression in terms of squared factors:
9y^2 + 18y + 16 - 4x^2 + 8x = 0
Rearranging the terms, we get:
(-4x^2 + 8x) + (9y^2 + 18y) + 16 = 0
Factoring out a negative one from the x terms and nine from the y terms:
-4(x^2 - 2x) + 9(y^2 + 2y) + 16 = 0
Now, we can complete the square for the x and y terms individually.
For the x terms:
1. Take half of the coefficient of x (-2) and square it: (-2/2)^2 = 1.
2. Add the result obtained in step 1 to both sides of the equation.
-4(x^2 - 2x + 1) + 9(y^2 + 2y) + 16 = 0
For the y terms:
1. Take half of the coefficient of y (2) and square it: (2/2)^2 = 1.
2. Add the result obtained in step 1 to both sides of the equation.
-4(x^2 - 2x + 1) + 9(y^2 + 2y + 1) + 16 = 0
Simplifying the equation, we get:
-4(x - 1)^2 + 9(y + 1)^2 + 16 = 0
Now, let's determine the type of conic:
The equation is in the form:
A(x - h)^2 + B(y - k)^2 = C
Since A is negative (-4) and B is positive (9), it is a hyperbola. Additionally, the equation can be written as:
[(x - 1)^2 / (4/(-1))] - [(y + 1)^2 / (4/9)] = 1
Therefore, the given equation is the standard form of a hyperbola.