Write the equation of the ellipse with center (2, - 3) vert , vertex and focus (2, 1) (2, - 3 + 2sqrt)
Why did the ellipse go to the therapist?
Because it couldn't focus on anything!
To find the equation of the ellipse, we need to determine the major and minor axes' lengths and the orientation of the ellipse.
1. The center of the ellipse is given as (h, k). From the given information, the center is (2, -3). So, h = 2 and k = -3.
2. The vertices of the ellipse are given as (h, k + a) and (h, k - a), where 'a' represents the distance from the center to a vertex along the major axis.
From the given information, the vertex points are (2, 1) and (2, -3 + 2√). We can see that the distance between the center and the vertex along the major axis is 1 - (-3 + 2√) = -3 + 2√ + 4 = 7 + 2√.
Therefore, a = (7 + 2√)/2 = 7/2 + √2.
3. The distance between the center and the focus along the major axis is given by c = √(a^2 - b^2), where 'b' represents the distance from the center to a co-vertex along the minor axis.
From the given information, the focus points are (2, 1) and (2, -3 + 2√).
We can see that the distance between the center and the focus along the major axis is 1 - (-3 + 2√) = 7 + 2√.
Therefore, c = 7 + 2√.
4. Using the equation of an ellipse, we have:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
Plugging in the values we found:
(x - 2)^2 / (7/2 + √2)^2 + (y + 3)^2 / b^2 = 1
Now, we can determine 'b' by using the relationship:
c = √(a^2 - b^2)
Reusing the value of 'a' and 'c' we found:
7 + 2√ = √((7/2 + √2)^2 - b^2)
Squaring both sides:
(7 + 2√)^2 = (7/2 + √2)^2 - b^2
49 + 28√ + 4 = 49/4 + 7√ + 2 + 4√2 - b^2
56 + 32√ = 49/4 + 7√ + 4√2 - b^2
Now, we can isolate 'b^2' on one side:
b^2 = 49/4 + 7√ + 4√2 - 56 - 32√
b^2 = 49/4 - 56 + 7√ - 32√ + 4√2
b^2 = 7/4 + 15√ + 4√2
Now, we can substitute this value of 'b^2' into the equation of the ellipse:
(x - 2)^2 / (7/2 + √2)^2 + (y + 3)^2 / (7/4 + 15√ + 4√2) = 1
Therefore, the equation of the ellipse is:
(x - 2)^2 / (49/4 + 14√2 + 2) + (y + 3)^2 / (7/4 + 15√ + 4√2) = 1
To write the equation of the ellipse with the given information, we can use the standard form of the equation for an ellipse:
(x - h)²/a² + (y - k)²/b² = 1,
where (h, k) represents the coordinates of the center of the ellipse, and "a" and "b" correspond to the semi-major and semi-minor axes lengths, respectively.
From the given information, we know that the center of the ellipse is (2, -3) and one of the vertices is (2, -3 + 2√2).
First, let's find the semi-major axis, "a". The semi-major axis is the distance from the center of the ellipse to a vertex. Using the coordinates of the center and the given vertex, we can calculate the distance using the distance formula:
a = √((x2 - x1)² + (y2 - y1)²),
where (x1, y1) = center = (2, -3) and (x2, y2) = vertex = (2, -3 + 2√2).
a = √((2 - 2)² + (-3 + 2√2 - (-3))²)
= √(0 + (2√2)²)
= √(0 + 4*2)
= √(8)
= 2√2.
Next, let's find the distance from the center to the focus. Using the coordinates of the center and one of the foci, we can calculate the distance:
c = √((x2 - x1)² + (y2 - y1)²),
where (x1, y1) = center = (2, -3) and (x2, y2) = focus = (2, -3 + 2√2).
c = √((2 - 2)² + (-3 + 2√2 - (-3))²)
= √(0 + 4*2)
= √(8)
= 2√2.
Since we know that the center of the ellipse is (h, k) = (2, -3), the values of "a" and "c", we can write the equation of the ellipse:
(x - 2)²/(2√2)² + (y + 3)²/ b² = 1.
Now, we just need to find the value of "b". Since we know the relationship between "a", "b", and "c" for an ellipse, we can use the formula:
c² = a² - b²,
where "c" is the distance from the center to the focus, and "a" and "b" are the semi-major and semi-minor axes lengths, respectively.
(2√2)² = (2√2)² - b²,
8 = 8 - b²,
b² = 0,
b = 0.
Since "b" is zero, the equation of the ellipse simplifies to:
(x - 2)²/(2√2)² + (y + 3)²/0² = 1,
(x - 2)²/(2√2)² + (y + 3)²/0 = 1,
(x - 2)²/(2√2)² = 1.
Therefore, the equation of the ellipse is:
(x - 2)²/(2√2)² + (y + 3)²/0 = 1,
(x - 2)²/(4√2) = 1.
This equation represents a degenerate ellipse, which is essentially a line segment with length "2a" (since "b" is zero) centered at (2, -3).
You have
a = 4
c = √2
so b = √12
The major axis is vertical, so
(x-2)^2/12 + (y+3)^2/16 = 1