With transverse axis parallel to the x-axis, center at (2,-2), passing through (2 + 3sqrt2, 0) and (2 + 3sqrt10, 4)
I assume you want the equation of a hyperbola
so we know:
(x-2)^2 /a^2 - (y+2)^2 /b^2 = 1
for point(2+3√2,0)
(3√2)^2 /a^2 - 2^2 /b^2 = 1
18/a^2 - 4/b^2 = 1
18b^2 - 4a^2 = a^2 b^2
for point(2+3√10,4)
(3√10)^2 /a^2 - 6^2/ b^2 = 1
90b^2 - 36a^2 = a^2b^2
so 90b^2 - 36a^2 = 18b^2 - 4a^2
72b^2 = 32a^
36b^2 = 16a^2 or b^2 = 16a^2 /36 = 4a^2 /9
6b = ± 4a
if 6b = 4a
b = 2a/3
sub b = 2a/3 into 18b^2 - 4a^2 = a^2b^2
18(4a^2/9) - 4a^2 = a^2(4a^2/9)
times 9
72a^2 - 36a^2 = 4a^4
4a^4 = 36a^2
divide both sides by 4a^2 , a ≠ 0
a^2 = 9
then b^2 = 4a^2 /9 = 9/9 = 1
(x-2)^2 /9 - (y+2)^2 = 1 is the equation
oops, messed up in the last few lines
the end should be:
a^2 = 9
then b^2 = 4a^2 /9 = 36/9 = 4
(x-2)^2 /9 - (y+2)^2/4 = 1 is the equation
To find the equation of an ellipse, we need the center of the ellipse and some information about its axes. In this case, we are given that the transverse axis is parallel to the x-axis and passes through two points. We can use this information to find the equation of the ellipse.
The center of the ellipse is given as (2, -2), which means the x-coordinate of the center is 2 and the y-coordinate of the center is -2.
The transverse axis is parallel to the x-axis, so the major axis of the ellipse is horizontal. This means the distance between the center and the two given points will give us the length of the major axis.
Let's calculate the length of the major axis using the distance formula. The distance formula between two points (x1, y1) and (x2, y2) is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For the first point, (2 + 3sqrt2, 0), the x-coordinate is 2 + 3sqrt2 and the y-coordinate is 0.
Using the distance formula, the distance between the center and the first point is:
d1 = sqrt((2 + 3sqrt2 - 2)^2 + (0 - (-2))^2)
d1 = sqrt((3sqrt2)^2 + 2^2)
d1 = sqrt(18 + 4)
d1 = sqrt(22)
For the second point, (2 + 3sqrt10, 4), the x-coordinate is 2 + 3sqrt10 and the y-coordinate is 4.
Using the distance formula, the distance between the center and the second point is:
d2 = sqrt((2 + 3sqrt10 - 2)^2 + (4 - (-2))^2)
d2 = sqrt((3sqrt10)^2 + 6^2)
d2 = sqrt(90 + 36)
d2 = sqrt(126)
The length of the major axis is the sum of these two distances:
2a = d1 + d2
Now, we can find the value of a:
2a = sqrt(22) + sqrt(126)
a = (sqrt(22) + sqrt(126))/2
To find the equation of the ellipse, we also need the distance between the center and the two foci. The distance between the center and each focus is given by the equation c = sqrt(a^2 - b^2), where b is the length of the minor axis.
Since we have the length of the major axis (2a), we need to find the length of the minor axis (2b). The minor axis is perpendicular to the major axis and passes through the center. We have two points on the minor axis to find its length: (-sqrt(22) + 2, -2) and (sqrt(22) + 2, -2).
Using the distance formula, we can find the length of the minor axis:
d3 = sqrt((-sqrt(22) + 2 - (sqrt(22) + 2))^2 + (-2 - (-2))^2)
d3 = sqrt((-sqrt(22))^2 + 0)
d3 = sqrt(22)
The length of the minor axis is d3. So, the length of the major axis is 2a = sqrt(22) + sqrt(126) and the length of the minor axis is 2b = sqrt(22).
Now, we can find the value of b:
2b = sqrt(22)
b = sqrt(22)/2
We can use the center and the value of a and b obtained to write the equation of the ellipse.
The equation of an ellipse with center (h, k) and semi-major axis a and semi-minor axis b is:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
In this case, the equation of the ellipse is:
(x - 2)^2/[(sqrt(22) + sqrt(126))/2]^2 + (y + 2)^2/(sqrt(22)/2)^2 = 1