The equation $\frac{(x-7)^2}{64}+\frac{(y+2)^2}{9}=1$ represents an ellipse. Which points are the vertices of the ellipse?
A. $(7,1)$ and $(7,-5)$
B. $(7,-10)$ and $(7,6)$
C. $(10,-2)$ and $(4,-2)$
D. $(15,-2)$ and $(-1,-2)$
Answer (2)
Identify the center of the ellipse as ( 7 , − 2 ) from the given equation.
Determine the semi-major axis a = 8 since the major axis is horizontal.
Calculate the vertices by adding and subtracting a from the x-coordinate of the center: ( 7 ± 8 , − 2 ) .
State the vertices as ( 15 , − 2 ) and ( − 1 , − 2 ) .
Explanation
Analyze the problem The equation of the ellipse is given by 64 ( x − 7 ) 2 + 9 ( y + 2 ) 2 = 1 . We need to find the coordinates of the vertices of this ellipse.
Recall the standard form of an ellipse The standard form of an ellipse centered at ( h , k ) is a 2 ( x − h ) 2 + b 2 ( y − k ) 2 = 1 if the major axis is horizontal, and b 2 ( x − h ) 2 + a 2 ( y − k ) 2 = 1 if the major axis is vertical, where a is the length of the semi-major axis and b is the length of the semi-minor axis. The vertices are located at a distance of a from the center along the major axis.
Identify the center and semi-major axis Comparing the given equation 64 ( x − 7 ) 2 + 9 ( y + 2 ) 2 = 1 with the standard form, we can identify the center of the ellipse as ( h , k ) = ( 7 , − 2 ) . Since 9"> 64 > 9 , the major axis is horizontal, and a 2 = 64 and b 2 = 9 . Thus, a = 64 = 8 and b = 9 = 3 .
Determine the coordinates of the vertices Since the major axis is horizontal, the vertices are located at ( h ± a , k ) . Substituting the values of h , k , and a , we get the vertices as ( 7 ± 8 , − 2 ) .
Calculate the coordinates of the vertices Calculating the coordinates, we have: Vertex 1: ( 7 + 8 , − 2 ) = ( 15 , − 2 ) Vertex 2: ( 7 − 8 , − 2 ) = ( − 1 , − 2 )
State the final answer Therefore, the vertices of the ellipse are ( 15 , − 2 ) and ( − 1 , − 2 ) .
Examples
Ellipses are commonly used in architecture and engineering. For example, the arches of bridges or the shape of whispering galleries can be elliptical. Knowing the vertices of an ellipse allows engineers to accurately design and construct these structures, ensuring stability and desired acoustic properties. Also, understanding ellipses is crucial in astronomy for describing the orbits of planets around stars.
The vertices of the ellipse represented by the equation are (15, -2) and (-1, -2). This was determined by identifying the center of the ellipse and the lengths of the semi-major and semi-minor axes. The correct answer is option D: (15, -2) and (-1, -2).
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