The <em>standard</em> form of the equation of the hyperbola that satisfies all conditions is (x - 2)²/4 - (y - 2)²/5 = 1 .
<h3>How to find the standard equation of a hyperbola</h3>
In this problem we must determine the equation of the hyperbola in its <em>standard</em> form from the coordinates of the foci and a <em>general</em> equation of a circle. Based on the location of the foci, we see that the axis of symmetry of the hyperbola is parallel to the x-axis. Besides, the center of the hyperbola is the midpoint of the line segment with the foci as endpoints:
(h, k) = 0.5 · (- 1, 2) + 0.5 · (5, 2)
(h, k) = (2, 2)
To determine whether it is possible that the vertices are endpoints of the diameter of the circle, we proceed to modify the <em>general</em> equation of the circle into its <em>standard</em> form.
If the vertices of the hyperbola are endpoints of the diameter of the circle, then the center of the circle must be the midpoint of the line segment. By algebra we find that:
x² + y² - 4 · x - 4 · y + 4= 0
(x² - 4 · x + 4) + (y² - 4 · y + 4) = 4
(x - 2)² + (y - 2)² = 2²
The center of the circle is the midpoint of the line segment. Now we proceed to determine the vertices of the hyperbola:
V₁(x, y) = (0, 2), V₂(x, y) = (4, 2)
And the distance from the center to any of the vertices is 2 (<em>semi-major</em> distance, a) and the semi-minor distance is:
b = √(c² - a²)
b = √(3² - 2²)
b = √5
Therefore, the <em>standard</em> form of the equation of the hyperbola that satisfies all conditions is (x - 2)²/4 - (y - 2)²/5 = 1 .
To learn more on hyperbolae: brainly.com/question/27799190
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