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
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.
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