For what values of a and b is the line 5x + y = b tangent to the parabola y = ax² when x = 4?

Solution:

Given the line: 5x + y = b.

et us find the slope of the line.

Step 1: Differentiate w.r.t ‘x’. 

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

-5 = 8a

 a = -5/8

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

y = (-5/8)(4)²

y = -10

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

5x + y = b

5(4) + -10 = b

b = 10

Thus for a = -5/8 and b = 10 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 5x + y = b tangent to the parabola y = ax² when x = 4?

Summary:

The line 5x + y = b tangent to the parabola y = ax² when x = 4 has values of a and b are -5/8 and 10 respectively.