Please help-What is the equation of the ellipse with foci (0,6), (0,-6) and co-vertices (2,0),(-2,0)
Please explain the steps because I have 5 to do for homework-Thank you-I'm really stuck on this
I thought the answer would be x^2/4 + y^2/40 = 1 but that can't be because the choices are:
x^2/1 + y^2/40 = 1 or x^2/1 + y^2/36
I'm really confused
F1(0,-6), F2(0,6).
The focus points show that the x-coordinate is constant @ zero, and the
y-coordinate varies from -6 to +6. Therefore, the focus points are on a vertical line. Which means we have a y-ellipse and a^2 is under y^2 in the Eq.
X^2/b^2 + Y^2/a^2 = 1.
b = +-2.
F^2 = a^2 - b^2,
F^2 = a^2 - 2^2 = 6^2,
a^2 - 4 = 36,
a^2 = 40,
Eq: X^2/4 + Y^2/40 = 1.
Check: Let Y = 0 and solve for X:
X^2/4 + 0/40 = 1,
X^2/4 = 1,
Multiply both sides by 4:
X^2 = 4,
X = +-2 = b = co-vertices = x-intercepts. In your 1st choice, the "1"
should be a 4.
NOTES:
1. "a" is always greater than b.
2. If this was an x-ellipse, a^2
would be located under X^2.
3. "a" is always on the major(longst)
axis.
To find the equation of the ellipse given the foci and co-vertices, follow these steps:
Step 1: Plot the given foci and co-vertices on a graph.
In this case, the foci are located at (0, 6) and (0, -6), and the co-vertices are at (2, 0) and (-2, 0). Draw the points on a graph to visualize the situation.
Step 2: Determine the center of the ellipse.
Since the foci are vertically aligned on the y-axis and the co-vertices are horizontally aligned on the x-axis, the center of the ellipse can be found at (0,0). The center represents the midpoint of the major axis.
Step 3: Determine the distance between the foci.
In this case, the distance between the foci is 2a, where a is the semi-major axis. Thus, 2a = 6 - (-6) = 12. Therefore, a = 6.
Step 4: Determine the distance between the co-vertices.
In this case, the distance between the co-vertices is 2b, where b is the semi-minor axis. Thus, 2b = 2 - (-2) = 4. Therefore, b = 2.
Step 5: Write the equation of the ellipse.
The standard equation for an ellipse centered at the origin is:
x^2/a^2 + y^2/b^2 = 1
Plugging in the values of a and b obtained from Steps 3 and 4, respectively, we get:
x^2/6^2 + y^2/2^2 = 1
which simplifies to:
x^2/36 + y^2/4 = 1
Therefore, the correct equation for the ellipse is x^2/36 + y^2/4 = 1.