Answer:
y = 1/2 x + 10.
Step-by-step explanation:
To find the slope of a tangent to the curve we find the derivative:
f(x) = (x^2 + 8) / (x - 2)
Using the quotient rule:
f'(x) = (x - 2)(2x) - (x^2 + 8) / (x - 2)^2
= (2x^2 - 4x - x^2 - 8) / (x - 2)^2
= (x^2 - 4x - 8) / (x - 2)^2
At x = 4 , the slope of the tangent
= (16 - 16 - 8) / 4
= -2.
So the slope of the normal = -1/-2 = 1/2.
The equation of the normal is
y = 1/2 x + c
When x = 4, y = (4^2 + 8) / 4 -2 = 24/2 = 12, so
12 = 1/2(4) + c
c = 12 - 2 = 10.
The required equation is y = 1/2 x + 10.
