The problem asks for values of $a$ and $b$ such that the line $y = -2x + 13$ is tangent to the graph of the function $f(x) = \frac{ax + b}{x - 1}$ at $x = 2$.

Steps to solve the problem:

1. Calculate $f(2)$:
   Since the line is tangent to the function at $x = 2$, the function $f(x)$ and the line $y = -2x + 13$ must intersect at this point.
   $f(2) = \frac{2a + b}{2 - 1} = 2a + b$ Since this point lies on the line $y = -2x + 13$, substituting $x = 2$ into the line equation: $y = -2(2) + 13 = -4 + 13 = 9$ Therefore: 2a + b = 9 \tag{1}

2. Find the derivative $f'(x)$:
   To ensure the line is tangent to the graph of the function, their slopes must also be equal at $x = 2$. First, compute the derivative of $f(x) = \frac{ax + b}{x - 1}$ using the quotient rule: $f'(x) = \frac{(a)(x - 1) - (ax + b)(1)}{(x - 1)^2} = \frac{ax - a - ax - b}{(x - 1)^2} = \frac{-a - b}{(x - 1)^2}$ At $x = 2$, the slope is: $f'(2) = \frac{-a - b}{(2 - 1)^2} = -a - b$ The slope of the tangent line is the same as the slope of the line $y = -2x + 13$, which is $-2$. Therefore: -a - b = -2 \tag{2}

3. Solve the system of equations:
   We now have the following system: 2a + b = 9 \tag{1} -a - b = -2 \tag{2} Solve Equation (2) for $a$ in terms of $b$: $-a = -2 + b \quad \Rightarrow \quad a = 2 - b$ Substitute into Equation (1): $2(2 - b) + b = 9$ $4 - 2b + b = 9$ $4 - b = 9 \quad \Rightarrow \quad b = -5$ Now, substitute $b = -5$ into the expression for $a$: $a = 2 - (-5) = 2 + 5 = 7$

4. Conclusion:
   The values of $a$ and $b$ are: $a = 7, \quad b = -5$

Thus, the correct values for $a$ and $b$ are not among the multiple-choice options provided in the image. However, based on calculations, these are the solutions ensuring the line is tangent at the given point.

Do you have any further questions or would you like more details?

Five related questions:

1. What is the geometric meaning of a tangent line to a graph?
2. How is the quotient rule applied to differentiate functions?
3. Why must the function value and derivative match at the tangency point?
4. Can a line be tangent to a function at multiple points? Why or why not?
5. How does changing $a$ and $b$ in the function $f(x) = \frac{ax + b}{x - 1}$ affect its graph?

Tip:

Always verify that both the point and the slope conditions are satisfied when determining tangency between a curve and a line.