What are the foci, center, vertices, major axis and minor axis of (x-2)29+(y-4)216=1?
oobleck, you didn't really answer the question though.
To find the foci, center, vertices, major axis, and minor axis of the ellipse given by the equation (x-2)²/9 + (y-4)²/216 = 1, we can use the standard form of an ellipse equation:
[(x-h)²/a²] + [(y-k)²/b²] = 1
where (h,k) represents the center of the ellipse, 'a' denotes the length of the major axis, 'b' represents the length of the minor axis, and c represents the distance from the center to each focus.
Comparing the given equation to the standard form, we can gather the following information:
Center: The center of the ellipse is given by the coordinates (h, k) = (2, 4).
Major Axis: The length of the major axis, denoted by '2a', is determined by 'a²'. In this case, a² = 9 → a = 3. Therefore, the length of the major axis is 2a = 2 * 3 = 6.
Minor Axis: The length of the minor axis, denoted by '2b', is determined by 'b²'. In this case, b² = 216 → b = √(216) = 6√6. Therefore, the length of the minor axis is 2b = 2 * 6√6 = 12√6.
Now, to find the foci, we need to calculate 'c', which is the distance from the center to each focus. The formula for calculating 'c' is given by:
c² = a² - b²
Substituting the values we found earlier, we can find 'c':
c² = 9 - 216 = -207
As the value under the square root is negative, we can conclude that this ellipse has imaginary foci. Implies there are no real foci for the given equation.
In summary, the information for the ellipse (x-2)²/9 + (y-4)²/216 = 1 is:
- Center: (2, 4)
- Major Axis: Length of 6 (from -3 to +3 on the x-axis, centered at 2)
- Minor Axis: Length of 12√6 (from -6√6 to +6√6 on the y-axis, centered at 4)
- Foci: This ellipse has no real foci.
you need to use things like ^ and / if you want to make any sense.
Again, here you have
a=4
b=3
c = √7
Now use what you know about a,b,c,h,k for ellipses. See your previous post.