Let the parabolas `y=x(c-x)a n dy=x^2+a x+b` touch each other at the point (1,0). Then (a) `a+b+c=0` (b) `a+b=2` (c) `b-c=1` (d) `a+c=-2`

Options

A. `a+b+c=0`
B. `a+b=2`
C. `b-c=1`
D. `a+c=-2`

Answer

To solve the problem, we need to determine the values of \( a \), \( b \), and \( c \) such that the two parabolas \( y = x(c - x) \) and \( y = x^2 + ax + b \) touch each other at the point \( (1, 0) \).

### Step 1: Find the slopes of both curves at the point of tangency

1. **Differentiate the first curve**:
\[ y_1 = x(c - x) \implies \frac{dy_1}{dx} = c - 2x \]
At the point \( (1, 0) \):
\[ \frac{dy_1}{dx} \bigg|_{x=1} = c - 2 \]

2. **Differentiate the second curve**:
\[ y_2 = x^2 + ax + b \implies \frac{dy_2}{dx} = 2x + a \]
At the point \( (1, 0) \):
\[ \frac{dy_2}{dx} \bigg|_{x=1} = 2 + a \]

### Step 2: Set the slopes equal at the point of tangency

Since the curves touch each other at \( (1, 0) \), their slopes must be equal:
\[ c - 2 = 2 + a \quad \text{(Equation 1)} \]

### Step 3: Set the equations equal at the point of tangency

Since both curves pass through the point \( (1, 0) \):
\[ 0 = 1(c - 1) \quad \text{(from the first curve)} \]
This simplifies to:
\[ c - 1 = 0 \implies c = 1 \quad \text{(Equation 2)} \]

### Step 4: Substitute \( c \) into Equation 1

Substituting \( c = 1 \) into Equation 1:
\[ 1 - 2 = 2 + a \implies -1 = 2 + a \implies a = -3 \quad \text{(Equation 3)} \]

### Step 5: Substitute \( a \) and \( c \) into the equation for the second curve

Now, we substitute \( a = -3 \) and \( c = 1 \) into the equation for the point \( (1, 0) \) in the second curve:
\[ 0 = 1^2 + (-3)(1) + b \implies 0 = 1 - 3 + b \implies b = 2 \quad \text{(Equation 4)} \]

### Step 6: Summarize the values

We have found:
- \( a = -3 \)
- \( b = 2 \)
- \( c = 1 \)

### Step 7: Check the given options

Now we check the options:

1. **Option (a)**: \( a + b + c = 0 \)
\[ -3 + 2 + 1 = 0 \quad \text{(True)} \]

2. **Option (b)**: \( a + b = 2 \)
\[ -3 + 2 = -1 \quad \text{(False)} \]

3. **Option (c)**: \( b - c = 1 \)
\[ 2 - 1 = 1 \quad \text{(True)} \]

4. **Option (d)**: \( a + c = -2 \)
\[ -3 + 1 = -2 \quad \text{(True)} \]

### Conclusion

The correct statements are:
- (a) \( a + b + c = 0 \)
- (c) \( b - c = 1 \)
- (d) \( a + c = -2 \)