For what values of a and b is the line 4x + y = b tangent to the parabola y = ax² when x = 4?

Solution:

Given the line 4x + y = b.

let us find the slope of the line

Step 1: Differentiate w.r.t ‘x’.

dy/dx = - 4 -------->(1)

Given parabola : y = ax²

Let us find the slope of parabola.

Step 2: Differentiate w.r.t ‘x’.

dy/ dx = d/dx (ax²)

dy/ dx = 2ax

At x = 4

dy/ dx = 8a --------> (2)

Step 3: Equate equation (1) and (2), we get

- 4 = 8a

a = - 4/ 8

a = -1/ 2

Step 4: Substitute the values of a and x in the equation of parabola to get y.

y = (-1/ 2) (4)²

y = - 8

Step 5: Substitute the values of a, x and y in the equation of line.

4x + y = b

4(4) + (- 8) = b

b = 8

Thus for a = -1/2 and b = 8 the line 4x + y = b is a tangent to the parabola y = ax²

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For what values of a and b is the line 4x + y = b tangent to the parabola y = ax² when x = 4?

Summary:

The line 4x + y = b tangent to the parabola y = ax² when x = 4 has values of a and b are -1/ 2 and 8 respectively.