Find the standard form equation for each ellipse described.
1) Major vertices at (0, 3) and (0, -3), minor vertices at (2, 0) and (-2, 0)
2) Major vertices at (7, 0) and (-7, 0), foci at (5, 0) and (-5, 0)
3) Minor vertices at (-2, -3) and (-2, -11), foci at (1, -7) and (-5, -7)
I'll do a freebie for you. Maybe you can attempt the rest and show your work.
The major semi-axis half the distance between the major vertices. Similarly for minor.
#1:
maj semi = 3
min semi = 2
x^2/4 + y^2/9 = 1
#3 is similar, but the center has been translated. Recall that the center at (h,k) means
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
To find the standard form equation for an ellipse, we will need to use some key information about the ellipse. The standard form equation for an ellipse centered at the origin is given by:
(x^2 / a^2) + (y^2 / b^2) = 1
If the ellipse is not centered at the origin, we will need to adjust the equation accordingly. Let's go through each problem step by step:
1) Major vertices at (0, 3) and (0, -3), minor vertices at (2, 0) and (-2, 0)
To find the value for a, which represents the semi-major axis, we need to find the distance between the major vertices.
Distance between major vertices = 2a = 6 (Since the y-coordinate of both vertices is 3 and -3)
Thus, a = 6/2 = 3.
To find the value for b, which represents the semi-minor axis, we need to find the distance between the minor vertices.
Distance between minor vertices = 2b = 4 (Since the x-coordinate of both vertices is 2 and -2)
Thus, b = 4/2 = 2.
Now we can substitute the values of a and b into the standard form equation:
(x^2 / 3^2) + (y^2 / 2^2) = 1
Simplifying further gives us the standard form equation:
(x^2 / 9) + (y^2 / 4) = 1
2) Major vertices at (7, 0) and (-7, 0), foci at (5, 0) and (-5, 0)
Since the major vertices are situated along the x-axis, the center of the ellipse is at (0,0). The value for a is the distance from the center to one of the major vertices:
a = 7 (The x-coordinate of the major vertices)
To find the value for c, which represents the distance from the center to each focus, we need to use the given information that the foci are located at (5, 0) and (-5, 0):
c = 5
To find the value for b, we can use the relationship between a, b, and c in an ellipse, which is given by:
c^2 = a^2 - b^2
Substituting the known values, we have:
5^2 = 7^2 - b^2
Simplifying further gives:
25 = 49 - b^2
Rearranging the equation, we have:
b^2 = 49 - 25 = 24
Taking the square root of both sides, we find:
b = √24 = 2√6
Now we can substitute the values of a and b into the standard form equation:
(x^2 / 7^2) + (y^2 / (2√6)^2) = 1
(x^2 / 49) + (y^2 / 24) = 1
3) Minor vertices at (-2, -3) and (-2, -11), foci at (1, -7) and (-5, -7)
Similarly to the previous example, since the minor vertices are situated along the y-axis, the center of the ellipse is at (-2, -7). The value for b is the distance from the center to one of the minor vertices:
b = 4 (The difference between the y-coordinates of the minor vertices)
To find the value for c, which represents the distance from the center to each focus, we can use the given information that the foci are located at (1, -7) and (-5, -7):
c = 5
To find the value for a, we can use the relationship between a, b, and c in an ellipse, which is given by:
c^2 = a^2 - b^2
Substituting the known values, we have:
5^2 = a^2 - 4^2
25 = a^2 - 16
Rearranging the equation, we have:
a^2 = 25 + 16 = 41
Taking the square root of both sides, we find:
a = √41
Now we can substitute the values of a and b into the standard form equation, while adjusting for the center of the ellipse:
((x + 2)^2 / (√41)^2) + ((y + 7)^2 / 4^2) = 1
((x + 2)^2 / 41) + ((y + 7)^2 / 16) = 1
And that's how we find the standard form equation for each ellipse described.