Final answer:
The value of a is 0.4 and the value of b is 10 for the line 4x - y = b to be tangent to the parabola y = ax² at x = 5.
Step-by-step explanation:
To find the values of a and b for which the line 4x - y = b is tangent to the parabola y = ax² at x = 5, we need to ensure that the line and the parabola meet at a single point and have the same slope at that point. At x = 5, the slope of the parabola, which is given by the derivative of y with respect to x, is 2ax.
By substituting x = 5 into the slope equation, we get the slope as 10a. The slope of the given line is 4, hence setting these equal gives us 10a = 4, so a = 0.4.
To find b, we substitute x = 5 into the original line equation and set it equal to the value of y at x = 5 in the parabola equation, which gives us 20 - y = b and y = 25a. Replacing a with 0.4 and solving for b, we obtain b = 20 - 10 = 10.