9x^2 + 16y^2 - 36x + 96y + 36 = 0 How do i find the major axis?
complete the square ...
9x^2 + 16y^2 - 36x + 96y + 36 = 0
9(x^2 - 4x + ...) + 16(y^2 + 6y + ..) = -36
9(x^2 - 4x + 4) + 16(y^2 + 6y + 9) = -36 +36+144
9(x-2)^2 + 16(y+3)^2 = 144
(x-2)^2/16 + (y+3)^2/9 = 1
so isn't a=4 and b=3 ?
To find the major axis of the given equation, we need to rewrite it in standard form, which is:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1, if a^2 > b^2 (ellipse)
or
((x - h)^2 / b^2) + ((y - k)^2 / a^2) = 1, if a^2 < b^2 (ellipse)
Looking at the given equation, we can see that the coefficients of x^2 and y^2 are both positive, so it represents an ellipse. Therefore, the standard form for the given equation is:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
Comparing this with the given equation, we can deduce the values of h, k, a, and b to find the major axis:
1. Rearrange the equation to group the x-terms together and the y-terms together:
9x^2 - 36x + 16y^2 + 96y = -36
2. Complete the square for x and y by dividing the coefficient of x by 2, squaring it, and adding it to both sides of the equation. Also, divide the coefficient of y by 2, square it, and add it to both sides:
9(x^2 - 4x) + 16(y^2 + 6y) = -36 + 36 + 96
Rewrite the x-terms and y-terms as perfect square binomials:
9(x^2 - 4x + 4) + 16(y^2 + 6y + 9) = -36 + 36 + 96
Simplify:
9(x - 2)^2 + 16(y + 3)^2 = 96
3. Divide both sides by 96 to put the equation in the standard form:
((x - 2)^2 / (96 / 9)) + ((y + 3)^2 / (96 / 16)) = 1
Simplify the fractions:
((x - 2)^2 / (32/3)) + ((y + 3)^2 / 6) = 1
4. Compare the equation with the standard form:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
We can deduce the following values:
h = 2
k = -3
a^2 = 32/3
b^2 = 6
Since a^2 is larger than b^2, the major axis is determined by the x-values, so the major axis is horizontal.
Therefore, the major axis of the ellipse is parallel to the x-axis.
To find the major axis of an ellipse represented by the given equation, you need to rewrite the equation in standard form. The standard form of an ellipse equation is:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
where (h, k) are the coordinates of the center of the ellipse, a represents the distance from the center to the major axis (the semi-major axis), and b represents the distance from the center to the minor axis (the semi-minor axis).
In your given equation, start by rearranging terms to group the x-terms and y-terms separately:
9x^2 - 36x + 16y^2 + 96y + 36 = 0
Next, complete the square for the x-terms and y-terms separately.
For the x-terms:
- Take the coefficient of x (which is 9) and divide it by 2, then square that value: (9/2)^2 = 81/4
- Add this value to both sides of the equation: 9x^2 - 36x + 81/4 + 16y^2 + 96y + 36 = 81/4
- Rearrange the equation: 9x^2 - 36x + 16y^2 + 96y + 81/4 + 36 = 81/4
- Combine constants: 9x^2 - 36x + 16y^2 + 96y + 117/4 = 81/4
Repeat the same process for the y-terms:
- Take the coefficient of y (which is 16) and divide it by 2, then square that value: (16/2)^2 = 64
- Add this value to both sides of the equation: 9x^2 - 36x + 16y^2 + 96y + 117/4 + 64 = 81/4 + 64
- Rearrange the equation: 9x^2 - 36x + 16y^2 + 96y + 181/4 = 145/4
Now, rewrite the equation in standard form:
- Divide both sides of the equation by the constant on the right side: (9x^2 - 36x + 16y^2 + 96y + 181/4) / (145/4) = 1
- Simplify: (4/145)(9x^2 - 36x + 16y^2 + 96y + 181) = 1
- Multiply both sides by 145/4 to eliminate the fraction: 9x^2 - 36x + 16y^2 + 96y + 181 = (145/4)
Now you can identify the values of a, b, h, and k by comparing the equation to the standard form:
- The value under x^2 is (9/81), which simplifies to 1/9. Therefore, a^2 = 1/9, so a = 1/3.
- The value under y^2 is (16/81), which simplifies to 4/9. Therefore, b^2 = 4/9, so b = 2/3.
- The x-coordinate of the center is given by h = -(-36)/(2*9) = 2.
- The y-coordinate of the center is given by k = -96/(2*16) = -3.
Now that you have the values of a, b, h, and k, you can determine the major axis of the ellipse. The major axis is the line passing through the center of the ellipse and has a length of 2a. In this case, the major axis has a length of 2 * (1/3) = 2/3.
Therefore, the major axis of the given ellipse is a horizontal line passing through the point (2, -3) and has a length of 2/3.