Answer:
The equation represents a hyperbola with a center at (5, 7), vertices at (14, 7), (-4, 7), (5, 20), and (5, -6), and an opening along the x-axis. (Ginny, AI)
Explanation:
The equation given represents a hyperbola. A hyperbola is a type of conic section, which is a curve formed by the intersection of a plane and a double cone. In the given equation, the x-term is squared and divided by 9^2, and the y-term is squared and divided by 13^2. The equation is set equal to 1.
To understand the hyperbola represented by this equation, we can look at its key features. The general form of a hyperbola equation is (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1, where (h, k) is the center of the hyperbola and a and b are the distances from the center to the vertices.
In this specific equation, the center is at (5, 7) because the terms (x - 5) and (y - 7) are squared and divided by 9^2 and 13^2 respectively.
The distance from the center to the vertices along the x-axis is a = 9. Therefore, the vertices are located at (5 ± 9, 7), which gives us the vertices (14, 7) and (-4, 7).
The distance from the center to the vertices along the y-axis is b = 13. Therefore, the vertices are located at (5, 7 ± 13), which gives us the vertices (5, 20) and (5, -6).
These vertices determine the shape of the hyperbola. In this case, the hyperbola opens horizontally because the x-term is squared and the y-term is not. The vertices (14, 7) and (-4, 7) are the farthest points on the hyperbola along the x-axis, while the vertices (5, 20) and (5, -6) are the farthest points along the y-axis.
In conclusion, the equation represents a hyperbola with a center at (5, 7), vertices at (14, 7), (-4, 7), (5, 20), and (5, -6), and an opening along the x-axis.