Final answer:
To find the values of a and b for the tangent line 2x - y = b to the parabola y = ax² at x = 3, we set up an equation for intersection and a condition for equal slopes. It turns out a must be 1/3 and b must be 3 for the line to be tangent at that point.
Step-by-step explanation:
The student is asking for the values of a and b such that the line 2x - y = b is tangent to the parabola y = ax² at the point where x = 3. To solve this, we must ensure two conditions are met. First, the line and the parabola must intersect at x = 3. Second, they must have the same slope at this point, since a tangent line should have a slope equal to the derivative of the parabola at that point.
To find the intersection, we substitute x = 3 into the parabola's equation to get the y-value at this point, which will be 9a. We then plug x = 3 and y = 9a into the line's equation to find the value of b, equating b to 6-9a. Next, to ensure the slopes are equal, we differentiate the parabola's equation with respect to x to find its slope 2ax, and then substitute x = 3 to get 6a, which must equal the slope of the line, which is 2. Thus, a must be 1/3.
Combining these results, we find that a = 1/3 and b = 3, so the line 2x - y = 3 would be tangent to the parabola y = (1/3)x² at the point where x = 3.