For what values of a and b is the line 2x + y = b tangent to the parabola y = ax² when x = 5?

Solution:

Given the line: 2x + y = b.

let us find the slope of the line

Step 1; Differentiate w.r.t ‘x’. 

dy/dx = - 2 --------> (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 = 5

dy/ dx = 10a --------> (2)

Step 3: Equate both the equations (1) and (2), we get

- 2 = 10a

a = -2/ 10

a = - 1/ 5

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

y = (-1/ 5) (5)²

y = - 5

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

2x + y = b

2(5) + - 5 = b

b = 5

Thus for a = -1/5 and b = 5 the line 2x + y = b is a tangent to the parabola y = ax²

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For what values of a and b is the line 2x + y = b tangent to the parabola y = ax² when x = 5?

Summary:

The line 2x + y = b tangent to the parabola y = ax² when x = 5 has values of a and b are -1/ 5 and 5 respectively.