Key Concepts and Formulas

• Point on a Curve: If a point $(x_0, y_0)$ lies on a curve $y = f(x)$, then $y_0 = f(x_0)$.
• Derivative as Slope of Tangent: The derivative $\frac{dy}{dx}$ of a curve $y = f(x)$ represents the slope of the tangent line to the curve at any point $(x, y)$.
• Slope-Intercept Form: The equation $y = mx + c$ represents a line with slope $m$ and y-intercept $c$.

Step-by-Step Solution

Step 1: Use the point (1, 2) to form an equation.

• Why: Since the curve passes through (1, 2), we can substitute these values into the equation of the curve to get a relationship between $a$, $b$, and $c$.
• Math: Substituting $x = 1$ and $y = 2$ into $y = ax^2 + bx + c$, we get: $2 = a(1)^2 + b(1) + c$ $a + b + c = 2 \quad \ldots (1)$

Step 2: Use the tangent line at the origin to find another equation.

• Why: The fact that the tangent line at the origin is $y = x$ gives us two pieces of information: the curve passes through (0, 0) and the slope of the tangent at (0, 0) is 1.
• Math (2a): Since the curve passes through (0, 0), we substitute $x = 0$ and $y = 0$ into the equation of the curve: $0 = a(0)^2 + b(0) + c$ $c = 0 \quad \ldots (2)$

Step 3: Find the derivative of the curve.

• Why: The derivative will give us the slope of the tangent line at any point on the curve.
• Math: Differentiating $y = ax^2 + bx + c$ with respect to $x$, we get: $\frac{dy}{dx} = 2ax + b$

Step 4: Use the slope of the tangent at the origin.

• Why: We know the tangent line at the origin is $y = x$, which has a slope of 1. We also know that the derivative at $x=0$ gives the slope of the tangent at the origin. Setting these equal gives us another equation.
• Math: The slope of the tangent line $y = x$ is 1. The derivative at $x = 0$ is: $\left.\frac{dy}{dx}\right|_{x=0} = 2a(0) + b = b$ Therefore, $b = 1 \quad \ldots (3)$

Step 5: Solve for a, b, and c.

• Why: We now have a system of three equations with three unknowns, which we can solve.
• Math: Substituting $c = 0$ and $b = 1$ into equation (1), we get: $a + 1 + 0 = 2$ $a = 1$ Therefore, $a = 1, b = 1, c = 0$.

Common Mistakes & Tips

• Remember that the tangent line at the origin implies the curve also passes through the origin.
• The slope of the line $y = x$ is 1.
• Double-check your differentiation and substitutions to avoid errors.

Summary

We used the given information that the curve passes through (1, 2) and has a tangent line $y = x$ at the origin to determine the values of $a$, $b$, and $c$. By substituting the point (1, 2) into the curve's equation, we obtained the equation $a + b + c = 2$. The tangent line information implied that the curve passes through (0, 0), leading to $c = 0$, and that the derivative at $x = 0$ is 1, leading to $b = 1$. Substituting these values back into the first equation, we found $a = 1$. Thus, $a = 1$, $b = 1$, and $c = 0$.

Final Answer The final answer is \boxed{a = 1, b = 1, c = 0}, which corresponds to option (B).