Which of the following is the equation of an ellipse centered at $(4,-1)$ having a horizontal minor axis of length 14 and a major axis of length 20?
A. $\frac{(x+1)^2}{49}+\frac{(y-4)^2}{100}=1$
B. $\frac{(x-4)^2}{100}+\frac{(y+1)^2}{49}=1$
C. $\frac{(x-4)^2}{14}+\frac{(y+1)^2}{20}=1$
D. $\frac{(x-4)^2}{49}+\frac{(y+1)^2}{100}=1$
Answer (2)
The ellipse is centered at ( 4 , − 1 ) , so h = 4 and k = − 1 .
The semi-minor axis is b = 2 14 = 7 , thus b 2 = 49 .
The semi-major axis is a = 2 20 = 10 , thus a 2 = 100 .
The equation of the ellipse is 49 ( x − 4 ) 2 + 100 ( y + 1 ) 2 = 1 .
Explanation
Analyze the problem The problem provides the center, the length of the horizontal minor axis, and the length of the major axis of an ellipse. We need to find the equation of the ellipse from the given options.
Recall the general equation of an ellipse The general equation of an ellipse centered at ( h , k ) with a horizontal minor axis is given by: b 2 ( x − h ) 2 + a 2 ( y − k ) 2 = 1 where:
( h , k ) is the center of the ellipse,
2 b is the length of the horizontal minor axis,
2 a is the length of the major axis.
Calculate the semi-minor and semi-major axes Given:
Center ( h , k ) = ( 4 , − 1 ) ,
Length of the horizontal minor axis 2 b = 14 , so b = 2 14 = 7 , and b 2 = 7 2 = 49 ,
Length of the major axis 2 a = 20 , so a = 2 20 = 10 , and a 2 = 1 0 2 = 100 .
Substitute the values into the equation Substitute the values of h , k , b 2 , and a 2 into the equation of the ellipse: 49 ( x − 4 ) 2 + 100 ( y − ( − 1 ) ) 2 = 1 49 ( x − 4 ) 2 + 100 ( y + 1 ) 2 = 1
Compare with the given options Comparing the derived equation with the given options, we find that option D matches the equation of the ellipse. 49 ( x − 4 ) 2 + 100 ( y + 1 ) 2 = 1
Examples
Ellipses are commonly used in architecture and engineering to design arches and bridges. For example, if an architect wants to design an elliptical arch for a bridge with a specific width and height, they can use the equation of an ellipse to determine the exact shape and dimensions of the arch. Knowing the center, major axis, and minor axis, they can create a structurally sound and aesthetically pleasing design.
The equation of the ellipse centered at (4, -1) with a horizontal minor axis of length 14 and major axis of length 20 is given by 49 ( x − 4 ) 2 + 100 ( y + 1 ) 2 = 1 . The correct choice among provided options is option B.
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