Question 1190880: Write the equation of the ELLIPSE that satisfies the given conditions
a. Vertex (6, 3), foci (4.3) and (4,3)
Found 3 solutions by CPhill, ikleyn, josgarithmetic:
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to find the equation of the ellipse:
1. **Find the Center:** The center of the ellipse is the midpoint of the segment connecting the foci.
Center = ((4 + 4)/2, (3 + 3)/2) = (4, 3)
2. **Determine the Orientation:** Since the foci have the same y-coordinate, the major axis is horizontal.
3. **Find 'c' (Distance from Center to Focus):** The distance from the center (4, 3) to a focus (4, 3) is c = |4-4| = 0. This would mean the foci and center are the same point, which means the ellipse is actually a circle. However, there is also a vertex at (6,3). Thus, it is an ellipse. The distance from the center (4,3) to the focus (4,3) is 0. The distance from the center (4,3) to the vertex (6,3) is 2, so a=2.
4. **Find 'a' (Semi-major Axis):** The semi-major axis is the distance from the center to a vertex.
a = distance from (4, 3) to (6, 3) = |6 - 4| = 2
5. **Find 'b' (Semi-minor Axis):** We use the relationship c² = a² - b² for ellipses. In this case, we must have the focus at a different location. Since the vertex is at (6,3) and the foci are at (4,3) and (4,3) we can assume that the foci are actually at (2,3) and (6,3). Thus c=2.
c² = a² - b²
2² = 2² - b²
4 = 4 - b²
b² = 0
This result b=0 would mean this is a line segment, not an ellipse. The foci must be different points.
The problem must have a typo. If the foci are at (2,3) and (6,3), and the vertex is at (6,3) then we have:
Center = (4,3)
a = 2
c = 2
c^2 = a^2 - b^2
4 = 4 - b^2
b = 0
If the foci are at (2,3) and (6,3) and a vertex is at (0,3) then we have:
Center = (4,3)
a = 4
c = 2
c^2 = a^2 - b^2
4 = 16 - b^2
b^2 = 12
b = sqrt(12) = 2sqrt(3)
6. **Write the Equation:** The general equation of an ellipse centered at (h, k) with a horizontal major axis is:
(x - h)² / a² + (y - k)² / b² = 1
Substitute the values we found (h = 4, k = 3, a = 4, b = 2√3):
(x - 4)² / 4² + (y - 3)² / (2√3)² = 1
Simplify:
(x - 4)² / 16 + (y - 3)² / 12 = 1
Therefore, if the foci are at (2,3) and (6,3), and a vertex is at (0,3) then the equation of the ellipse is (x - 4)²/16 + (y - 3)²/12 = 1.
Answer by ikleyn(52847)  (Show Source):
You can put this solution on YOUR website! .
As the problem is printed in the post, it is FATALLY DEFECTIVE.
It says that one focus is (4.3), but it determines one number, not two numbers - coordinates of the focus.
Even if (4.3) should be read as (4,3), then it means that two foci coincide,
so the ellipse is, actually, a circle.
So, the way, how the problem is presented, creates ambiguity and confuses a reader.
Answer by josgarithmetic(39625)  (Show Source):
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