If the curve `y=ax^(2)+bx+c` passes through the point (1, 2) and the line y = x touches it at the origin, then

Options

A. a = 1, b = -1, c = 0
B. a = 1, b = 1, c = 0
C. a = -1, b = 1, c = 0
D. none of these

Answer

To solve the problem, we need to determine the values of \( a \), \( b \), and \( c \) for the quadratic curve \( y = ax^2 + bx + c \) given that it passes through the point \( (1, 2) \) and the line \( y = x \) is tangent to the curve at the origin.

### Step-by-Step Solution:

1. **Substituting the Point (1, 2) into the Curve Equation**:
Since the curve passes through the point \( (1, 2) \), we can substitute \( x = 1 \) and \( y = 2 \) into the equation:
\[ 2 = a(1)^2 + b(1) + c \]
This simplifies to:
\[ a + b + c = 2 \quad \text{(Equation 1)} \]

**Hint**: Remember to substitute the coordinates directly into the equation of the curve.

2. **Finding the Derivative of the Curve**:
The derivative of the curve \( y = ax^2 + bx + c \) gives us the slope of the tangent at any point \( x \):
\[ \frac{dy}{dx} = 2ax + b \]

**Hint**: The derivative represents the slope of the tangent line to the curve at any point.

3. **Setting Up the Tangent Condition at the Origin**:
Since the line \( y = x \) is tangent to the curve at the origin \( (0, 0) \), we need to evaluate the derivative at \( x = 0 \):
\[ \frac{dy}{dx} \bigg|_{x=0} = 2a(0) + b = b \]
The slope of the line \( y = x \) is \( 1 \). Therefore, we have:
\[ b = 1 \quad \text{(Equation 2)} \]

**Hint**: The slope of the tangent line at the point of tangency must equal the slope of the line itself.

4. **Substituting \( b \) into Equation 1**:
Now that we have \( b = 1 \), we can substitute this value back into Equation 1:
\[ a + 1 + c = 2 \]
Simplifying this gives:
\[ a + c = 1 \quad \text{(Equation 3)} \]

**Hint**: Substitute known values into equations to simplify and solve for unknowns.

5. **Finding the Value of \( c \) Using the Origin**:
Since the curve passes through the origin \( (0, 0) \), we can substitute \( x = 0 \) and \( y = 0 \) into the curve equation:
\[ 0 = a(0)^2 + b(0) + c \]
This simplifies to:
\[ c = 0 \quad \text{(Equation 4)} \]

**Hint**: Check the curve equation at the point of tangency to find additional values.

6. **Finding the Value of \( a \)**:
Now substituting \( c = 0 \) into Equation 3:
\[ a + 0 = 1 \]
Thus:
\[ a = 1 \]

**Hint**: Use the values you have found to solve for the remaining unknowns.

7. **Final Values**:
We have determined:
\[ a = 1, \quad b = 1, \quad c = 0 \]

### Conclusion:
The values of \( a \), \( b \), and \( c \) are:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 0 \)