What is the major axis and the domain of the equation 9x^2 + 16y^2 - 36x + 96y + 36 = 0 ?
To determine the major axis and domain of the equation 9x^2 + 16y^2 - 36x + 96y + 36 = 0, we should rewrite the equation in the standard form of an ellipse equation.
Step 1: Move the constant term to the other side of the equation:
9x^2 - 36x + 16y^2 + 96y = -36
Step 2: Group the x-terms together and the y-terms together:
(9x^2 - 36x) + (16y^2 + 96y) = -36
Step 3: Factor out the common coefficient from each grouped term:
9(x^2 - 4x) + 16(y^2 + 6y) = -36
Step 4: Complete the square for both the x and y terms by adding the square of half the coefficient of x and y, respectively, inside the parentheses:
9(x^2 - 4x + 4) + 16(y^2 + 6y + 9) = -36 + 36 + 144
Simplifying gives:
9(x - 2)^2 + 16(y + 3)^2 = 144
Step 5: Divide both sides of the equation by the constant term on the right side:
[(x - 2)^2]/16 + [(y + 3)^2]/9 = 1
Comparing this equation to the standard form of an ellipse: (x-h)^2/a^2 + (y-k)^2/b^2 = 1
where (h, k) represents the center, a represents the semi-major axis, and b represents the semi-minor axis, we can determine the values for the major axis and the domain.
In this case, we have:
Center (h, k) = (2, -3)
Semi-major axis a = 4 (since a^2 = 16, so a = √16 = 4)
Semi-minor axis b = 3 (since b^2 = 9, so b = √9 = 3)
The major axis refers to the longest diameter of the ellipse, which is twice the length of the semi-major axis. In this case, the major axis is 2a = 2(4) = 8.
The domain of the ellipse is the range of possible x-values. Since the ellipse is symmetric with respect to the x-axis and its center is (2, -3), the domain is given by h-a ≤ x ≤ h+a, or 2-4 ≤ x ≤ 2+4, which simplifies to -2 ≤ x ≤ 6.
Thus, the major axis has a length of 8 and the domain of the equation is -2 ≤ x ≤ 6.
To find the major axis and the domain of the equation 9x^2 + 16y^2 - 36x + 96y + 36 = 0, we can start by rearranging the equation:
9x^2 + 16y^2 - 36x + 96y + 36 = 0
We can rewrite it as:
9x^2 - 36x + 16y^2 + 96y = -36
Next, we can complete the square for both the x and y terms.
For the x terms:
9(x^2 - 4x) + 16y^2 + 96y = -36
To complete the square for the x terms, we need to add (4/2)^2 = 4 to the expression inside the parenthesis:
9(x^2 - 4x + 4) + 16y^2 + 96y = -36 + 9(4)
Simplifying further:
9(x - 2)^2 + 16y^2 + 96y = -36 + 36
Next, let's complete the square for the y terms:
9(x - 2)^2 + 16(y^2 + 6y + 9) = 0
To complete the square for the y terms, we need to add (6/2)^2 = 9 to the expression inside the parenthesis:
9(x - 2)^2 + 16(y^2 + 6y + 9) = 0 + 16(9)
Simplifying further:
9(x - 2)^2 + 16(y + 3)^2 = 144
Now, we can rewrite the equation in standard form:
(x - 2)^2/16 + (y + 3)^2/9 = 1
Comparing this equation to the standard form of an ellipse, (x - h)^2/a^2 + (y - k)^2/b^2 = 1, we can see that the major axis is along the y-axis and the minor axis is along the x-axis.
Thus, the major axis is vertical.
The domain of the equation is all real numbers, as there are no restrictions on the value of x in this equation.