The equation [tex]$\frac{(y-2)^2}{64}+\frac{x^2}{9}=1$[/tex] represents an ellipse. Which lines represent the approximate directrices of the ellipse? Round to the nearest tenth.
A. [tex]$x=-8.6$[/tex] and [tex]$x=8.6$[/tex]
B. [tex]$x=-6.6$[/tex] and [tex]$x=10.6$[/tex]
C. [tex]$y=-8.6$[/tex] and [tex]$y=8.6$[/tex]
D. [tex]$y=-6.6$[/tex] and [tex]$y=10.6[/tex]
Answer (2)
Identify the center ( 0 , 2 ) , semi-major axis a = 8 , and semi-minor axis b = 3 from the ellipse equation.
Calculate the eccentricity e = 8 55 using the formula e = 1 − a 2 b 2 .
Determine the distance from the center to the directrices e a = 55 64 ≈ 8.6 .
Find the equations of the directrices: y = 2 ± 8.6 , which gives y = − 6.6 and y = 10.6 . The final answer is y = − 6.6 and y = 10.6 .
Explanation
Problem Analysis We are given the equation of an ellipse: 64 ( y − 2 ) 2 + 9 x 2 = 1 . Our goal is to find the equations of the directrices of this ellipse, rounded to the nearest tenth.
Identify Ellipse Parameters The equation is in the standard form a 2 ( y − k ) 2 + b 2 ( x − h ) 2 = 1 , where ( h , k ) is the center of the ellipse, a is the semi-major axis, and b is the semi-minor axis. In our case, the center is ( 0 , 2 ) , a 2 = 64 , and b 2 = 9 . Thus, a = 64 = 8 and b = 9 = 3 .
Calculate Eccentricity Since b"> a > b , the major axis is along the y-axis. The eccentricity e of the ellipse is given by the formula e = 1 − a 2 b 2 . Substituting the values of a and b , we get: e = 1 − 64 9 = 64 64 − 9 = 64 55 = 8 55 .
Distance to Directrices The distance from the center to the directrices is given by e a . Substituting the values of a and e , we get: e a = 8 55 8 = 55 8 ⋅ 8 = 55 64 .
Rationalize and Approximate To rationalize the denominator, we multiply the numerator and denominator by 55 :
55 64 = 55 64 55 .
Now, we approximate the value: 55 64 55 ≈ 55 64 ⋅ 7.416 ≈ 55 474.624 ≈ 8.630 .
Find Directrices Equations Since the major axis is along the y-axis, the equations of the directrices are given by y = k ± e a , where ( 0 , k ) is the center of the ellipse, so k = 2 . Therefore, the equations of the directrices are: y = 2 ± 8.630 .
Calculating the two values: y 1 = 2 + 8.630 ≈ 10.6 y 2 = 2 − 8.630 ≈ − 6.6 .
Final Answer Therefore, the equations of the directrices are approximately y = − 6.6 and y = 10.6 .
Conclusion The approximate directrices of the ellipse are y = − 6.6 and y = 10.6 .
Examples
Ellipses and their properties, including directrices, are used in various fields such as astronomy (planetary orbits), optics (designing lenses), and architecture (designing elliptical arches). For example, understanding the directrices of an elliptical orbit helps predict the closest and farthest points a planet reaches in its orbit around a star. In architecture, elliptical shapes are sometimes used for aesthetic reasons, and knowing the directrices can aid in the precise construction of these shapes.
The current of 15.0 A flowing for 30 seconds delivers a charge of 450 C. This charge corresponds to approximately 2.81 x 10^21 electrons flowing through the device. The calculation involves using the relationship between current, time, and the charge of an electron.
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