Key Concepts and Formulas

• X-intercepts: A point where a curve intersects the x-axis has a y-coordinate of 0.
• Derivative as Slope: The derivative of a function $y = f(x)$, denoted as $\frac{dy}{dx}$, represents the slope of the tangent line to the curve at a given point.
• Equation of a Tangent Line: The equation of the tangent line to the curve $y = f(x)$ at the point $(x_0, y_0)$ is given by $y - y_0 = m(x - x_0)$, where $m = f'(x_0)$ is the slope of the tangent at that point.

Step-by-Step Solution

Step 1: Find the x-intercepts of the curve

We are given the curve $y = x^2 - 3x + 2$. To find the x-intercepts, we set $y = 0$ and solve for $x$: $x^2 - 3x + 2 = 0$ $(x - 1)(x - 2) = 0$ Thus, the x-intercepts are $x = 1$ and $x = 2$. The points where the curve intersects the x-axis are $(1, 0)$ and $(2, 0)$.

Step 2: Find the derivative of the curve

The derivative of $y = x^2 - 3x + 2$ with respect to $x$ is: $\frac{dy}{dx} = 2x - 3$ This derivative gives the slope of the tangent line to the curve at any point $x$.

Step 3: Find the slopes of the tangents at the x-intercepts

At $x = 1$, the slope of the tangent is: $m_1 = 2(1) - 3 = -1$ At $x = 2$, the slope of the tangent is: $m_2 = 2(2) - 3 = 1$

Step 4: Find the equations of the tangent lines

At the point $(1, 0)$, the equation of the tangent line is: $y - 0 = -1(x - 1)$ $y = -x + 1$ $x + y = 1$

At the point $(2, 0)$, the equation of the tangent line is: $y - 0 = 1(x - 2)$ $y = x - 2$ $x - y = 2$

Step 5: Match the tangent lines with the given lines

We are given the lines $x + y = a$ and $x - y = b$. Comparing $x + y = 1$ with $x + y = a$, we get $a = 1$. Comparing $x - y = 2$ with $x - y = b$, we get $b = 2$.

Step 6: Calculate the ratio a/b

We need to find the value of $\frac{a}{b}$: $\frac{a}{b} = \frac{1}{2}$

Common Mistakes & Tips

• Sign Errors: Be careful with signs when calculating the derivative and forming the equation of the tangent line.
• Incorrect Factoring: Ensure that you factor the quadratic equation correctly to find the x-intercepts. Double-check your factorization.
• Confusing a and b: Keep track of which tangent line corresponds to which given equation.

Summary

We found the x-intercepts of the curve, calculated the derivatives at those points to find the slopes of the tangent lines, and then determined the equations of the tangent lines. By comparing these equations with the given lines $x + y = a$ and $x - y = b$, we found the values of $a$ and $b$ and finally calculated the ratio $\frac{a}{b}$.

The final answer is \boxed{1/2}.