Find the length of the axes of the ellipse defined by the equation shown below. If necessary, round to the nearest tenth.
16x^2 + 9y^2 + 96x - 36y + 36 = 0
Length of Major Axis
Length of Minor Axis:
can you please recalculate that answer is wrong
wrong again
the bot managed it up to
16(x^2 + 6x) + 9(y^2 - 4y) = -36
then .. it should be
16(x^2 + 6x + 9) + 9(y^2 - 4y + 4) = -36 + 144 + 36
16(x + 3)^2 + 9(y - 2)^2 = 144
divide everybody by 144
(x + 3)^2 / 9 + (y - 2)^2 / 16 = 1
now it is easy, do the rest in your head
I apologize for my previous errors. Here is the corrected solution:
We start with the equation:
16x^2 + 9y^2 + 96x - 36y + 36 = 0
We complete the square for both x and y terms:
16(x^2 + 6x) + 9(y^2 - 4y) = -36
16(x^2 + 6x + 9) + 9(y^2 - 4y + 4) = -36 + 16(9) + 9(4)
16(x + 3)^2 + 9(y - 2)^2 = 144
Dividing both sides by 144, we get:
(x + 3)^2/9 + (y - 2)^2/16 = 1.
This is the standard form of an ellipse:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
where (h,k) is the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis.
Comparing this to our equation, we see that the center of the ellipse is (-3,2). Also, a^2 = 16, so a = 4, and b^2 = 9, so b = 3.
Therefore, the length of the major axis is 2a = 8, and the length of the minor axis is 2b = 6. Rounded to the nearest tenth, we have:
Length of Major Axis = 8.0
Length of Minor Axis = 6.0
Find the foci of the ellipse defined by the equation shown below. If necessary, round to the nearest tenth.
9x^2 + 25y^2 - 54x + 150y + 81 =0
To find the foci of the ellipse, we first need to rewrite the equation in standard form:
9x^2 + 25y^2 - 54x + 150y + 81 = 0
Dividing both sides by 81, we get:
x^2/9 + y^2/3.24 - 2x/3 + 5y/1.62 + 1 = 0
Now, we can identify the values of a, b, and c, where a is the length of the semi-major axis, b is the length of the semi-minor axis, and c is the distance from the center to each focus:
a^2 = 9, so a = 3
b^2 = 3.24, so b ≈ 1.8
c^2 = a^2 - b^2 = 5.76 - 3.24 = 2.52, so c ≈ 1.59
Therefore, the distance from the center to each focus is approximately 1.59. Since the center of the ellipse is not given, we cannot determine the coordinates of the foci directly from the given equation. However, we know that the foci lie on the major axis, which is parallel to the x-axis and passes through the center.
If we assume that the center of the ellipse is (h, k), then the equation of the major axis is:
(y - k)^2/b^2 = 1 - (x - h)^2/a^2
When x = h ± a, we have:
(y - k)^2/b^2 = 0
y = k
This means that the foci lie on the line y = k, which passes through the center of the ellipse. Since the distance from the center to each focus is 1.59, the coordinates of the foci are:
(h - 1.59, k) and (h + 1.59, k)
Without additional information on the center of the ellipse, we cannot compute the exact coordinates of the foci.
In order to find the length of the axes, we need to rewrite the equation in standard form. We start by completing the square for both x and y terms:
16(x^2 + 6x) + 9(y^2 - 4y) = -36
We add and subtract the terms needed to complete the squares, making sure to balance the equation:
16(x^2 + 6x + 9) + 9(y^2 - 4y + 4) = -36 + 16(9) + 9(4)
Simplifying the right side, we get:
16(x + 3)^2 + 9(y - 2)^2 = 196
Dividing both sides by 196, we get:
(x + 3)^2/49 + (y - 2)^2/196 = 1.
This is the standard form of an ellipse:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
where (h,k) is the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis.
Comparing this to our equation, we see that the center of the ellipse is (-3,2). Also, a^2 = 196/16 = 49/4, so a = 7/2, and b^2 = 196/64 = 49/16, so b = 7/4.
Therefore, the length of the major axis is 2a = 7, and the length of the minor axis is 2b = 7/2. Rounded to the nearest tenth, we have:
Length of Major Axis = 7.0
Length of Minor Axis = 3.5
I apologize for the error in my previous calculation. Here is the corrected solution:
In order to find the length of the axes, we need to rewrite the equation in standard form. We start by completing the square for both x and y terms:
16x^2 + 9y^2 + 96x - 36y + 36 = 0
16(x^2 + 6x) + 9(y^2 - 4y) = -36
We add and subtract the terms needed to complete the squares, making sure to balance the equation:
16(x^2 + 6x + 9) - 16(9) + 9(y^2 - 4y + 4) - 9(4) = -36 - 16(9) - 9(4)
Simplifying the left side, we get:
16(x + 3)^2 + 9(y - 2)^2 = 100
Dividing both sides by 100, we get:
(x + 3)^2/25 + (y - 2)^2/11.1111... = 1.
This is the standard form of an ellipse:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
where (h,k) is the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis.
Comparing this to our equation, we see that the center of the ellipse is (-3,2). Also, a^2 = 25, so a = 5, and b^2 = 11.1111..., so b ≈ 3.33.
Therefore, the length of the major axis is 2a = 10, and the length of the minor axis is 2b ≈ 6.67. Rounded to the nearest tenth, we have:
Length of Major Axis = 10.0
Length of Minor Axis = 6.7