In the xy-plane, a circle has the equation (x - 2)^2 + (y - 3)^2 = 25. A line is tangent to the circle at the point (5,…

Correct answer: C. 43/4

The correct answer is 43/4. The equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. For the given circle, the center is (2, 3). The slope of the radius from the center (2, 3) to the point of tangency (5, 7) is (7 - 3) / (5 - 2) = 4/3. A line tangent to a circle is perpendicular to the radius at the point of tangency. The slope of the tangent line is the negative reciprocal of the slope of the radius, which is -3/4. Using the point-slope form of a linear equation, y - y1 = m(x - x1), with the point (5, 7) and slope -3/4, the equation of the tangent line is y - 7 = (-3/4)(x - 5). To find the y-intercept, set x = 0: y - 7 = (-3/4)(-5), which simplifies to y - 7 = 15/4. Adding 7 to both sides gives y = 15/4 + 7 = 15/4 + 28/4 = 43/4. Therefore, the y-intercept is 43/4. A is incorrect because it is the y-intercept of a line passing through (5, 7) with a slope of 4/3, the same slope as the radius, not the perpendicular slope of the tangent line. B is incorrect. This value results from using the positive reciprocal slope (3/4) instead of the negative reciprocal (-3/4). It can also result from a sign error when multiplying (-3/4) by (-5). D is incorrect because it is the x-intercept of the tangent line, found by setting y=0 in the line's equation, not the y-intercept.