For what values of a and b is the line 4x + y = b tangent to the parabola y = ax² when x = 3?

Solution:

Given the line 4x + y = b.

let us find the slope of the line

Step 1: Differentiate w.r.t ‘x’.

Step 1: Differentiate w.r.t ‘x’.

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

Slope of parabola is y = ax²

Step 2: Differentiate w.r.t ‘x’.

dy/ dx = d/dx (ax²)

dy/ dx = 2ax

At x = 3

dy/ dx = 6a --------> (2)

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

- 4 = 6a

a = - 4/ 6

a = - 2/ 3

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

y = (- 2/3) (3)²

y = - 6

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

4x + y = b

4(3) + - 6 = b

b = 6

Thus for a = -2/3 and b = 6 the line 4x + y = b is a tangent to the parabola y = ax²

---

For what values of a and b is the line 4x + y = b tangent to the parabola y = ax² when x = 3?

Summary:

The line 4x + y = b tangent to the parabola y = ax² when x = 3 has values of a and b are -2/ 3 and 6 respectively.