Which of the following equations represents an ellipse with a minor axis of length 10 and foci located at $(3,6)$ and $(7,6)$?
A. $\frac{(x-5)^2}{25}+\frac{(y-6)^2}{29}=1$
B. $\frac{(x+5)^2}{29}+\frac{(y+6)^2}{25}=1$
C. $\frac{(x-5)^2}{29}+\frac{(y-6)^2}{25}=1$
D. $\frac{(x-5)^2}{21}+\frac{(y-6)^2}{25}=1
Answer (2)
Find the center of the ellipse as the midpoint of the foci: ( 5 , 6 ) .
Determine b from the minor axis length: b = 5 .
Determine c from the distance between the foci: c = 2 .
Calculate a 2 using a 2 = b 2 + c 2 : a 2 = 29 .
Write the equation of the ellipse: 29 ( x − 5 ) 2 + 25 ( y − 6 ) 2 = 1 .
Explanation
Find the parameters of the ellipse. The ellipse has a minor axis of length 10, which means that 2 b = 10 . Therefore, b = 2 10 = 5 . The foci are located at ( 3 , 6 ) and ( 7 , 6 ) . The center of the ellipse is the midpoint of the foci, which is ( 2 3 + 7 , 2 6 + 6 ) = ( 5 , 6 ) . The distance between the foci is 2 c = ∣7 − 3∣ = 4 , so c = 2 4 = 2 . Since a 2 = b 2 + c 2 , we have a 2 = 5 2 + 2 2 = 25 + 4 = 29 . The major axis is horizontal because the foci have the same y-coordinate. The equation of the ellipse is of the form a 2 ( x − h ) 2 + b 2 ( y − k ) 2 = 1 where ( h , k ) is the center.
Write the equation of the ellipse. Now we can plug in the values we found for the center ( h , k ) = ( 5 , 6 ) , a 2 = 29 , and b 2 = 25 into the equation of the ellipse:
Substitute the values. 29 ( x − 5 ) 2 + 25 ( y − 6 ) 2 = 1
Compare with the options. Comparing this equation with the given options, we see that it matches option C.
Final Answer. Therefore, the equation of the ellipse is 29 ( x − 5 ) 2 + 25 ( y − 6 ) 2 = 1 .
Examples
Ellipses are commonly used in architecture and engineering to design arches and bridges. For example, if you were designing an elliptical archway with a specific height (related to the minor axis) and the location of its focal points, you would use the equation of an ellipse to determine the overall shape and dimensions of the arch. This ensures the structure is both aesthetically pleasing and structurally sound.
The equation of the ellipse with a minor axis of length 10 and foci located at (3,6) and (7,6) is 29 ( x − 5 ) 2 + 25 ( y − 6 ) 2 = 1 , which corresponds to option C. This is derived by first finding the center, the lengths of the axes, and using the standard form of the ellipse. Thus, the correct choice is C.
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