Final answer:
To find the values of a and b that make the line 3x - y = b tangent to the parabola y = ax² when x = 4, set the slopes of the line and parabola equal to each other and solve for a. Substitute the value of a into the equation of the line and set it equal to b to find the range of values for b. The tangent line and parabola will be tangent when a and b satisfy the equation 12 - 16a² = b.
Step-by-step explanation:
To find the values of a and b for which the line 3x - y = b is tangent to the parabola y = ax² when x = 4, we need to equate the slopes of the line and the parabola at the point of tangency. The slope of the line can be found by rearranging the equation to y = 3x - b, which is in the form of y = mx + c, where m is the slope. Therefore, the slope of the line is 3. The slope of the parabola at any point is given by the derivative of y = ax² with respect to x, which is 2ax. Setting the slopes equal to each other, we have 3 = 2a(4), which simplifies to 12a = 3 and a = 1/4. Substituting the value of a into the equation of the line, we have 3x - y = b, or 3x - ax² = b. Substituting x = 4, we get 3(4) - a(4)² = b, which simplifies to 12 - 16a² = b. Therefore, for the line and parabola to be tangent, the values of a and b must satisfy the equation 12 - 16a² = b.