A hyperbola is defined by the equation (y-4)^2/25 - (x-8)^2 = 1
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To answer step-by-step, let's analyze the given equation of the hyperbola:
(y - 4)^2/25 - (x - 8)^2 = 1
Step 1: Identify the center of the hyperbola.
The center of a hyperbola is represented by the values (h, k) in the form (x-h)^2/a^2 - (y-k)^2/b^2 = 1. In this case, the center is (8, 4).
Step 2: Determine the orientation of the hyperbola.
To determine the orientation, we look at the signs in the equation. Since the positive term (y-4)^2 appears before the negative term (x-8)^2, we can conclude that the hyperbola opens vertically. So, it is a vertically-oriented hyperbola.
Step 3: Identify the major and minor axes.
In a vertically-oriented hyperbola, the major axis is the vertical line that passes through the center, and the minor axis is the horizontal line perpendicular to the major axis.
In this case, the major axis is the line passing through (8, 4), which means it is parallel to the y-axis.
The minor axis is the line perpendicular to the major axis, and in this case, it is parallel to the x-axis.
Step 4: Find the vertices.
The distance between the center and the vertices is given by the square root of the denominator in the equation. For a vertically-oriented hyperbola, the vertices lie on the major axis.
In this case, the square root of 25 is 5. So, the distance between the center and the vertices is 5 units.
Since the center is (8, 4), the coordinates of the vertices are (8, 4 + 5) = (8, 9) and (8, 4 - 5) = (8, -1).
Step 5: Find the co-vertices.
The distance between the center and the co-vertices is given by the square root of the numerator in the equation. For a vertically-oriented hyperbola, the co-vertices lie on the minor axis.
In this case, the square root of 1 is 1. So, the distance between the center and the co-vertices is 1 unit.
Since the center is (8, 4), the coordinates of the co-vertices are (8 + 1, 4) = (9, 4) and (8 - 1, 4) = (7, 4).
Step 6: Determine the foci.
The distance between the center and the foci is given by the square root of a^2 + b^2, where a and b are the lengths of the semi-major and semi-minor axes, respectively.
In this case, the square root of (25 + 1) = sqrt(26), which is approximately 5.1. So, the distance between the center and the foci is approximately 5.1 units.
Since the center is (8, 4), the coordinates of the foci are (8, 4 + 5.1) ≈ (8, 9.1) and (8, 4 - 5.1) ≈ (8, -1.1).
Step 7: Draw a graph of the hyperbola.
Using the information obtained from the previous steps, plot the center, vertices, co-vertices, and foci on a graph to sketch the hyperbola.
That's it! You have now determined the properties of the hyperbola defined by the equation (y-4)^2/25 - (x-8)^2 = 1.