Find the value of a so that the tangent line to y=x^2 -2(sqrtx) +1

Question

is perpendicular to the line ay+2x=2?

Answer 1

The function y = x² - 2√x + 1

Differentiating  on each side with respect of x .

y' = 2x - 2(1/2√x)

y' = 2x - 1/√x

Here y' is slope of the tangent line.

The tangent line is perpendicular to  ay + 2x = 2.

Write the equation in slope intercept form y = mx + b.

y = (- 2/a)x + 2/a

Compare the above equation with y = mx + b.

Slope  is - 2/a.

We know that perpendicular lines slopes are negative reciprocal to each other.

Therefore slope of the tangent line is a/2.

2x - 1/√x  = a/2

2(2x - 1/√x ) = a

a = 4x - 2/√x .