The sum of the ordinates of point of contacts of the common tangent to the parabolas `y = x^2 + 4x +8` and `y=x^2+8x +4` is

Answer

To find the sum of the ordinates of the points of contact of the common tangent to the parabolas \( y = x^2 + 4x + 8 \) and \( y = x^2 + 8x + 4 \), we will follow these steps:

### Step 1: Rewrite the equations of the parabolas in vertex form
1. **First parabola**:
\[ y = x^2 + 4x + 8 \]
Completing the square:
\[ y = (x^2 + 4x + 4) + 4 = (x + 2)^2 + 4 \]
So, the vertex form is \( y = (x + 2)^2 + 4 \).

2. **Second parabola**:
\[ y = x^2 + 8x + 4 \]
Completing the square:
\[ y = (x^2 + 8x + 16) - 12 = (x + 4)^2 - 12 \]
So, the vertex form is \( y = (x + 4)^2 - 12 \).

### Step 2: Set up the equations for the common tangent
Assuming the common tangent has the equation \( y = mx + c \), we need to find \( m \) and \( c \) such that it is tangent to both parabolas.

### Step 3: Find the conditions for tangency
For the first parabola:
\[ mx + c = x^2 + 4x + 8 \]
Rearranging gives:
\[ x^2 + (4 - m)x + (8 - c) = 0 \]
For this to have a double root (tangency), the discriminant must be zero:
\[ (4 - m)^2 - 4(8 - c) = 0 \]
This simplifies to:
\[ (4 - m)^2 = 32 - 4c \quad \text{(1)} \]

For the second parabola:
\[ mx + c = x^2 + 8x + 4 \]
Rearranging gives:
\[ x^2 + (8 - m)x + (4 - c) = 0 \]
For this to have a double root, the discriminant must also be zero:
\[ (8 - m)^2 - 4(4 - c) = 0 \]
This simplifies to:
\[ (8 - m)^2 = 16 - 4c \quad \text{(2)} \]

### Step 4: Solve the system of equations
From equations (1) and (2):
1. \( (4 - m)^2 = 32 - 4c \)
2. \( (8 - m)^2 = 16 - 4c \)

Setting them equal to each other:
\[ 32 - 4c = 16 - 4c + 16 - 8m + m^2 \]
This simplifies to:
\[ 32 = 32 - 8m + m^2 \]
Thus, we have:
\[ 8m = m^2 \implies m(m - 8) = 0 \]
So, \( m = 0 \) or \( m = 8 \).

### Step 5: Find the corresponding \( c \)
1. If \( m = 0 \):
\[ 4c = 32 \implies c = 8 \]
The tangent line is \( y = 8 \).

2. If \( m = 8 \):
Substitute \( m = 8 \) into either equation:
\[ (4 - 8)^2 = 32 - 4c \implies 16 = 32 - 4c \implies 4c = 16 \implies c = 4 \]
The tangent line is \( y = 8x + 4 \).

### Step 6: Find the points of contact
1. For \( y = 8 \):
- Substitute into the first parabola:
\[ 8 = x^2 + 4x + 8 \implies x^2 + 4x = 0 \implies x(x + 4) = 0 \implies x = 0, -4 \]
- The points of contact are \( (0, 8) \) and \( (-4, 8) \).

2. For \( y = 8x + 4 \):
- Substitute into the first parabola:
\[ 8x + 4 = x^2 + 4x + 8 \implies x^2 - 4x + 4 = 0 \implies (x - 2)^2 = 0 \implies x = 2 \]
- The point of contact is \( (2, 20) \).

### Step 7: Sum of the ordinates
- The ordinates of the points of contact are \( 8 \) and \( 20 \).
- Therefore, the sum of the ordinates is:
\[ 8 + 20 = 28 \]

### Final Answer
The sum of the ordinates of the points of contacts of the common tangent to the parabolas is **28**.