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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 = 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.
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