Final answer:
To find the values of the constant c for which the line 2y = x + c is a tangent to the curve y = (2x + 6)/x, we need to set the derivative of the curve equal to the slope of the line. After solving the resulting equation, we find that x = -6. Substituting this value back into the equation of the line, we find that c = 4.
Step-by-step explanation:
In order for the line 2y = x + c to be a tangent to the curve y = (2x + 6)/x, the slopes of the line and the curve must be equal at the point of tangency. To find this, we can set the derivative of the curve equal to the slope of the line, which is 1/2. The derivative of the curve is given by:
y' = (-6 - 2x) / x^2
Setting this equal to 1/2, we get:
(-6 - 2x) / x^2 = 1/2
After solving this equation, we find that x = -6. Substituting this value back into the equation of the curve, we get y = (2(-6) + 6)/(-6), which simplifies to y = -1. The values of c for which the line is a tangent to the curve are the values that make y = -1 when substituting x = -6.
Substituting x = -6 into the equation of the line, we get 2(-1) = -6 + c. Simplifying this equation, we find that c = 4. Therefore, the correct answer is A. c = 4.