For what values of a and b is the line 3x + y = b tangent to the parabola y = ax² when x = 2?

Solution:

Given the line: 3x + y = b.

let us find the slope of the line.

Step 1: Differentiate w.r.t ‘x’. 

dy/dx = - 3 --------> (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 = 2

dy/ dx = 4a --------> (2)

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

- 3 = 4a

a = - 3/ 4

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

y = (- 3/ 4) (2)²

y = - 3

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

3x + y = b

3(2) + - 3 = b

b = 3

Thus for a = -3/4 and b = 3 the line 3x + y = b is a tangent to the parabola y = ax²

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For what values of a and b is the line 3x + y = b tangent to the parabola y = ax² when x = 2?

Summary:

The line 3x + y = b tangent to the parabola y = ax² when x = 2 has values of a and b are -3/ 4 and 3 respectively.