The equation $\frac{(y-8)^2}{49}+\frac{(x-1)^2}{36}=1$ represents an ellipse. Which points are the approximate locations of the foci of the ellipse? Round to the nearest tenth.
A. $(-8,-4.6)$ and $(-8,2.6)$
B. ( $-2.6,-8$ ) and ( $4.6,8$ )
C. $(1,4.4)$ and $(1,11.6)$
D. $(11.6,1)$ and $(4.4,1)$
Answer (1)
Identify the center of the ellipse as ( 1 , 8 ) , the semi-major axis a = 7 , and the semi-minor axis b = 6 .
Calculate c using the formula c = a 2 − b 2 = 13 ≈ 3.6 .
Determine the coordinates of the foci using the formula ( 1 , 8 ± c ) .
The approximate locations of the foci are ( 1 , 4.4 ) and ( 1 , 11.6 ) .
Explanation
Identify the center, semi-major axis, and semi-minor axis. The equation of the ellipse is given by 49 ( y − 8 ) 2 + 36 ( x − 1 ) 2 = 1 .
First, we identify the center of the ellipse as ( h , k ) = ( 1 , 8 ) .
Next, we determine the semi-major axis a and the semi-minor axis b . Since 36"> 49 > 36 , the major axis is along the y-axis. Thus, a 2 = 49 and b 2 = 36 . Therefore, a = 49 = 7 and b = 36 = 6 .
Calculate the distance from the center to each focus. To find the foci, we need to calculate c , where c 2 = a 2 − b 2 .
Substituting the values of a and b , we get: c 2 = 7 2 − 6 2 = 49 − 36 = 13
Therefore, c = 13 .
Approximate c and determine the location of the foci. We approximate the value of c to the nearest tenth. Since 13 ≈ 3.6055 , we have c ≈ 3.6 .
Since the major axis is along the y-axis, the foci are located at ( h , k ± c ) , where ( h , k ) is the center of the ellipse.
Thus, the coordinates of the foci are ( 1 , 8 + 3.6 ) and ( 1 , 8 − 3.6 ) .
Calculate the coordinates of the foci. Calculating the coordinates of the foci: ( 1 , 8 + 3.6 ) = ( 1 , 11.6 ) ( 1 , 8 − 3.6 ) = ( 1 , 4.4 )
Therefore, the approximate locations of the foci of the ellipse are ( 1 , 4.4 ) and ( 1 , 11.6 ) .
Examples
Ellipses are commonly used in architecture and engineering to design arches and bridges. Understanding how to find the foci of an ellipse is crucial in these applications because the foci determine the shape and stability of the structure. For example, an elliptical arch can distribute weight evenly, making it stronger than a circular arch. By calculating the foci, engineers can optimize the design of these structures to ensure they are both aesthetically pleasing and structurally sound.
