If `m` be the slope of common tangent of `y = x^2 - x + 1` and `y = x^2 – 3x + 1`. Then `m` is equal to

Options

A. 16
B. 7
C. 9
D. none of these

Explanation

To find the slope \( m \) of the common tangent of the parabolas given by the equations \( y = x^2 - x + 1 \) and \( y = x^2 - 3x + 1 \), we can follow these steps:

### Step 1: Rewrite the equations in standard form
The first parabola is:
\[ y = x^2 - x + 1 \]
We can complete the square:
\[ y = \left(x - \frac{1}{2}\right)^2 + \frac{3}{4} \]
This shows that the vertex is at \( \left(\frac{1}{2}, \frac{3}{4}\right) \).

The second parabola is:
\[ y = x^2 - 3x + 1 \]
Completing the square gives:
\[ y = \left(x - \frac{3}{2}\right)^2 - \frac{5}{4} \]
This shows that the vertex is at \( \left(\frac{3}{2}, -\frac{5}{4}\right) \).

### Step 2: Write the equations of the tangents
The general equation of the tangent to a parabola \( y = ax^2 + bx + c \) at a point \( (x_0, y_0) \) is given by:
\[ y - y_0 = m(x - x_0) \]
For the first parabola, the equation of the tangent can be expressed as:
\[ y - \frac{3}{4} = m\left(x - \frac{1}{2}\right) - \frac{m^2}{4} \]
Rearranging gives:
\[ y = mx - \frac{m}{2} + \frac{3}{4} - \frac{m^2}{4} \]

For the second parabola, the equation of the tangent is:
\[ y + \frac{5}{4} = m\left(x - \frac{3}{2}\right) - \frac{m^2}{4} \]
Rearranging gives:
\[ y = mx - \frac{3m}{2} - \frac{5}{4} - \frac{m^2}{4} \]

### Step 3: Set the equations equal to find the slope
Since both tangents are common, we set the right-hand sides of both equations equal:
\[ mx - \frac{m}{2} + \frac{3}{4} - \frac{m^2}{4} = mx - \frac{3m}{2} - \frac{5}{4} - \frac{m^2}{4} \]
This simplifies to:
\[ -\frac{m}{2} + \frac{3}{4} = -\frac{3m}{2} - \frac{5}{4} \]
Now, simplify and solve for \( m \):
\[ -\frac{m}{2} + \frac{3}{4} + \frac{3m}{2} + \frac{5}{4} = 0 \]
Combining like terms:
\[ \frac{2m}{2} + \frac{8}{4} = 0 \]
This simplifies to:
\[ m + 2 = 0 \quad \Rightarrow \quad m = -2 \]

### Final Answer
Thus, the slope \( m \) of the common tangent is:
\[ \boxed{-2} \]