Find the standard form of the equation of the ellipse with the following characteristics.
foci: (+/-5, 0) major axis of length: 22
a = 22 / 2 = 11 = radius of major axis.
c^2 = a^2 - b^2.
5^2 = (11)^2 - b^2,
25 = 121 - b^2,
b^2 = 121 - 25,
b^2 = 96,
b = 4*sqrt(6).
STD Form: X^2 / a^2 + Y^2 / b^2 = 1.
X^2 / 121 + Y^2 / 96 = 1.
X^2 / (11)^2 + Y^2 / 4*sqrt(6),
To find the standard form of the equation for an ellipse, you need the coordinates of the foci and the length of the major axis. The standard form of the equation for an ellipse with a horizontal major axis is given by:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1 (1)
where (h,k) is the center of the ellipse, a is the distance from the center to a vertex along the major axis, and b is the distance from the center to a vertex along the minor axis.
In this case, the foci are (+/-5, 0) and the major axis has a length of 22. Since the foci lie on the x-axis, the center of the ellipse will be at the origin (0,0).
Step 1: Find the value of a:
The distance from the center to a vertex along the major axis is equal to half the length of the major axis. Thus, a = 22/2 = 11.
Step 2: Find the value of c:
For an ellipse, c represents the distance from the center to a focus. In this case, c = 5 (since the foci are located at (+/-5, 0)).
Step 3: Find the value of b:
Using the relationship between a, b, and c for an ellipse, we have c^2 = a^2 - b^2.
Substituting the known values, 5^2 = 11^2 - b^2.
Simplifying, 25 = 121 - b^2.
Rearranging the equation, we have b^2 = 121 - 25.
Simplifying further, we get b^2 = 96.
Taking the square root of both sides gives us b = √96 = 4√6.
Step 4: Substitute the known values into equation (1):
Plugging in the values from steps 1, 2, and 3 into equation (1), we have:
(x-0)^2/11^2 + (y-0)^2/(4√6)^2 = 1.
Simplifying further, the standard form of the equation for the given ellipse is:
x^2/121 + y^2/96 = 1.
To find the standard form of the equation of an ellipse with the given characteristics, we need to use the general equation for an ellipse and then manipulate it to match the given information.
The general equation for an ellipse centered at the origin with horizontal major axis is:
((x-h)^2/a^2) + ((y-k)^2/b^2) = 1
In this case, the foci are located at (+/-5, 0), and the major axis has a length of 22 units.
First, let's find the value of 'a', which represents the semi-major axis. The semi-major axis is half the length of the major axis. Therefore, a = 22/2 = 11.
Next, we can determine the value of 'c', which represents the distance from the center of the ellipse to each focus. In this case, c = 5.
We can then calculate the value of 'b', which is related to 'a' and 'c' using the equation c^2 = a^2 - b^2. Substituting the known values, we get 5^2 = 11^2 - b^2. Solving for 'b', we find that b^2 = 121 - 25, which gives us b^2 = 96.
Now, we can substitute the values of 'a' and 'b' into the general equation of an ellipse. The standard form for this ellipse is:
((x-0)^2/11^2) + ((y-0)^2/√96)^2 = 1
Simplifying this equation further, we have:
(x^2/121) + (y^2/96) = 1
Therefore, the standard form of the equation of the ellipse is:
x^2/121 + y^2/96 = 1