Sum of slopes of common tangent to `y = (x^(2))/(4) - 3x +10` and `y = 2 - (x^(2))/(4)` is (a) -6 (b) -3 (c) `1/2` (d) none of these

Options

A. `-6`
B. `-3`
C. `1//2`
D. none of these

Explanation

To find the sum of the slopes of the common tangents to the parabolas given by the equations \( y = \frac{x^2}{4} - 3x + 10 \) and \( y = 2 - \frac{x^2}{4} \), we can follow these steps:

### Step 1: Set Up the Common Tangent Equation
Assume the equation of the common tangent is given by:
\[ y = mx + c \]
where \( m \) is the slope and \( c \) is the y-intercept.

### Step 2: Substitute into the First Parabola
For the first parabola \( y = \frac{x^2}{4} - 3x + 10 \), substitute \( y \) from the tangent equation:
\[ mx + c = \frac{x^2}{4} - 3x + 10 \]
Rearranging gives:
\[ \frac{x^2}{4} - (3 + m)x + (10 - c) = 0 \]
This is a quadratic equation in \( x \).

### Step 3: Condition for Tangency
For the line to be a tangent, the discriminant of this quadratic must be zero:
\[ b^2 - 4ac = 0 \]
Here, \( a = \frac{1}{4} \), \( b = -(3 + m) \), and \( c = 10 - c \). Thus, we have:
\[ (3 + m)^2 - 4 \cdot \frac{1}{4} \cdot (10 - c) = 0 \]
Simplifying this gives:
\[ (3 + m)^2 - (10 - c) = 0 \]
\[ (3 + m)^2 + c - 10 = 0 \tag{1} \]

### Step 4: Substitute into the Second Parabola
Now substitute into the second parabola \( y = 2 - \frac{x^2}{4} \):
\[ mx + c = 2 - \frac{x^2}{4} \]
Rearranging gives:
\[ \frac{x^2}{4} + mx + (c - 2) = 0 \]
Again, for tangency, the discriminant must be zero:
\[ m^2 - 4 \cdot \frac{1}{4} \cdot (c - 2) = 0 \]
This simplifies to:
\[ m^2 - (c - 2) = 0 \]
\[ m^2 + 2 - c = 0 \tag{2} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( (3 + m)^2 + c - 10 = 0 \)
2. \( m^2 + 2 - c = 0 \)

From equation (2), we can express \( c \):
\[ c = m^2 + 2 \]
Substituting this into equation (1):
\[ (3 + m)^2 + (m^2 + 2) - 10 = 0 \]
Expanding and simplifying:
\[ (3 + m)^2 + m^2 - 8 = 0 \]
\[ 9 + 6m + m^2 + m^2 - 8 = 0 \]
\[ 2m^2 + 6m + 1 = 0 \]

### Step 6: Find the Sum of the Roots
Using the quadratic formula \( ax^2 + bx + c = 0 \), the sum of the roots \( m_1 + m_2 \) is given by:
\[ -\frac{b}{a} = -\frac{6}{2} = -3 \]

### Conclusion
Thus, the sum of the slopes of the common tangents is:
\[ \boxed{-3} \]