For what values of a and b is the line 3x + y = b tangent to the parabola y = ax^2 when x = 4?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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If the line 3x + y = b is tangential to y = ax^2 at x = 4, the slope of the line is equal to the value of the derivative of y = ax^2 at x = 4.

3x + y = b

=> y = -3x + b

The slope of the tangent is -3.

The derivative of y = ax^2 is 2ax. For x = 4, 2ax = 8a.

8a = -3

=> a = -3/8

At x = 4, y = (-3/8)*16 = -6

Substituting in 3x + y = b

=> 3*4 - 6 = b

=> 12 - 6 = b

=> b = 6

The value of a = -3/8 and b = 6

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