Final answer:
The curve is defined by the equation y = ax² + bx + c. From the conditions given we find c = 0 and b = 4. The value of a cannot be determined without an additional point.
Step-by-step explanation:
The student has provided information about a curve y = ax² + bx + c that needs to satisfy two conditions: it passes through the origin and is tangent to the line y = 4x at that point. To be tangent to the line at the origin, the slope of the curve at that point (which is the derivative of y with respect to x) must be equal to the slope of the line, which is 4. Using these conditions, we can find the coefficients a, b, and c.
Firstly, as the curve passes through the origin, when x = 0, y must also equal 0. This gives us our first equation: c = 0.
The derivative of y with respect to x is 2ax + b. Since the curve is tangent to y = 4x at the origin, the derivatives must be equal at x = 0, meaning that b = 4.
However, with the information given, we cannot find a specific value for a without an additional point that the curve passes through (which was not provided in the question). To fully determine the values of a, b, and c, we need at least one more condition or point.