Find the equation of thea.       Hyperbola with center (1, 1), vertex (1, 5), conjugate axis 6.b.      Circle with center on the line y = x, tangent to the line y = 5 and has radius 2.

a. The equation of a hyperbola with center (h, k), vertices (h, k ± a), and conjugate axis of length 2b is given by:

(y - k)²/a² - (x - h)²/b² = 1

Given that the center is (1, 1), the vertex is (1, 5), and the conjugate axis is 6, we can find the values of a and b. The distance from the center to the vertex gives us a, which is |5 - 1| = 4. The length of the conjugate axis is 2b, so b = 6/2 = 3.

Substituting these values into the equation, we get:

(y - 1)²/4² - (x - 1)²/3² = 1
Simplifying, we get the equation of the hyperbola:
(y - 1)²/16 - (x - 1)²/9 = 1

b. The equation of a circle with center (h, k) and radius r is given by:

(x - h)² + (y - k)² = r²

Given that the center lies on the line y = x, we know that h = k. The circle is tangent to the line y = 5, which means the distance from the center to this line is equal to the radius, which is 2. Therefore, the y-coordinate (and thus the x-coordinate) of the center is 5 - 2 = 3.

Substituting these values into the equation, we get:

(x - 3)² + (y - 3)² = 2²
Simplifying, we get the equation of the circle:
(x - 3)² + (y - 3)² = 4