If the lines `x-y=a` and `x+y=b` are tangents for `y=x^2-3x+2` then `a/b=`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) for the lines \( x - y = a \) and \( x + y = b \) that are tangents to the parabola \( y = x^2 - 3x + 2 \).

### Step 1: Find the vertex of the parabola
The given parabola can be rewritten in vertex form. The standard form of a parabola is \( y = ax^2 + bx + c \). Here, we have:
\[ y = x^2 - 3x + 2 \]
To find the vertex, we can use the formula for the x-coordinate of the vertex:
\[ x_v = -\frac{b}{2a} = -\frac{-3}{2 \cdot 1} = \frac{3}{2} \]
Now substituting \( x_v \) back into the equation to find \( y_v \):
\[ y_v = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) + 2 = \frac{9}{4} - \frac{9}{2} + 2 = \frac{9}{4} - \frac{18}{4} + \frac{8}{4} = -\frac{1}{4} \]
Thus, the vertex of the parabola is \( \left(\frac{3}{2}, -\frac{1}{4}\right) \).

### Step 2: Find the slopes of the tangent lines
For the line \( x - y = a \), we can rewrite it as:
\[ y = x - a \]
This line has a slope of \( 1 \).

For the line \( x + y = b \), we can rewrite it as:
\[ y = -x + b \]
This line has a slope of \( -1 \).

### Step 3: Find the points of tangency
To find the points of tangency, we need to set the derivative of the parabola equal to the slopes of the lines.

1. **For the line \( x - y = a \)**:
The derivative of the parabola \( y = x^2 - 3x + 2 \) is:
\[ \frac{dy}{dx} = 2x - 3 \]
Setting this equal to the slope of the line:
\[ 2x - 3 = 1 \implies 2x = 4 \implies x = 2 \]
Now substituting \( x = 2 \) back into the parabola to find \( y \):
\[ y = 2^2 - 3(2) + 2 = 4 - 6 + 2 = 0 \]
Thus, the point of tangency is \( (2, 0) \).

Now, substituting this point into the line equation:
\[ 2 - 0 = a \implies a = 2 \]

2. **For the line \( x + y = b \)**:
Setting the derivative equal to the slope:
\[ 2x - 3 = -1 \implies 2x = 2 \implies x = 1 \]
Now substituting \( x = 1 \) back into the parabola to find \( y \):
\[ y = 1^2 - 3(1) + 2 = 1 - 3 + 2 = 0 \]
Thus, the point of tangency is \( (1, 0) \).

Now, substituting this point into the line equation:
\[ 1 + 0 = b \implies b = 1 \]

### Step 4: Calculate \( \frac{a}{b} \)
Now that we have \( a = 2 \) and \( b = 1 \):
\[ \frac{a}{b} = \frac{2}{1} = 2 \]

### Final Answer
\[ \frac{a}{b} = 2 \]