In ellipse vertices, foci and center they lie in the same line and on the major axis.
Problem 1 :
Foci: (±5, 0); major axis of length 12
Solution:
Foci are F_{1 }(5, 0) and F_{2 }(-5, 0). By observing the given foci, the ellipse is symmetric about x-axis.
Length of major axis = 12
2a = 12
a = 6
Midpoint of foci = center
Here the foci are on the x-axis, so the major axis is along the x-axis.
So, the equation of the ellipse is
2a = 12
a = 6
a^{2} = 36
c = 5
b^{2} = a^{2} - c^{2}
b^{2} = 6^{2} - 5^{2}
b^{2} = 36 - 25
b^{2} = 11
Hence the required equation of ellipse is
Problem 2 :
Foci: (±2, 0); major axis of length 8
Solution:
Given the major axis is 8 and foci are (±2, 0).
Here the foci are on the x-axis, so the major axis is along the x-axis.
So, the equation of the ellipse is
2a = 8
a = 4
a^{2} = 16
c = 2
b^{2} = a^{2} - c^{2}
b^{2} = 4^{2} - 2^{2}
b^{2} = 16 - 4
b^{2} = 12
Hence the required equation of ellipse is
Problem 3 :
Foci: (0, 0), (4, 0); major axis of length 8
Solution:
The midpoint between the foci is the center
The distance between the foci is equal to 2c
The major axis length is equal to 2a
2a = 8
a = 4
b^{2} = a^{2} - c^{2}
= 4^{2} - 2^{2}
= 16 - 4
b^{2} = 12
The standard equation of an ellipse with a horizontal major axis is
Problem 4 :
Foci: (0, 0), (0, 8); major axis of length 16
Solution:
The midpoint between the foci is the center
The distance between the foci is equal to 2c
The major axis length is equal to 2a
2a = 16
a = 8
b^{2} = a^{2} - c^{2}
b^{2} = 8^{2} - 4^{2}
b^{2} = 64 - 16
b^{2} = 48
By observing foci, since x-coordinates are same. The ellipse is symmetric about y-axis.
Problem 5 :
Vertices: (0, 4), (4, 4); minor axis of length 2
Solution:
The center of the ellipse
By observing center and foci, the ellipse is symmetric about x-axis.
Length of minor axis = 2
2b = 2
b = 1
Length of the major axis
a^{2} = 16
Equation of the ellipse
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