What is the equation of the line normal to # f(x)=x/(x-2) # at # x=1#?

Question

What is the equation of the line normal to #  f(x)=x/(x-2) # at #  x=1#?

Evelyn Agard · Accepted Answer

Equation of normal is #x-2y-3=0# The line normal to #f(x)# at #x=x_0# is perpendicular to the tangent at the point #(x_0,f(x_0))#. As slope of the tangent at #x=x_0# is #f'(x_0)#, the slope of normal is #-1/(f'(x_0))# and equation of normal is #y-f(x_0)=-1/(f'(x_0))(x-x_0)#. Here we are seeking normal at #x=1# and as #f(x)=x/(x-2)#, at this point #f(1)=1/(1-2)=-1#. For slope let us work out derivative of #f(x)=x/(x-2)=1+2/(x-2)# and #f'(x)=-2/(x-2)^2# and slope of tangent at #x=1# is #f'(1)=-2/(1-2)^2=-2# The slope of normal is then #-1/-2=1/2# and equation of tangent is #y-(-1)=-2(x-1)# or #2x+y-1=0# and that of normal is #y-(-1)=1/2(x-1)# or #2y+2=x-1# or #x-2y-3=0# graph{(x-2y-3)(xy-x-2y)(2x+y-1)=0 [-10, 10, -5, 5]}

Aurora Alvarado · Answer

undefined The equation of the line normal to f(x)=x/(x-2) at x=1 is y = -3x + 4.

Ezekiel Alexander · Answer

undefined To find the equation of the line normal to $ f(x) = \frac{x}{x-2} $ at $ x = 1 $, we first need to find the derivative of the function $ f(x) $, and then find the slope of the tangent line at $ x = 1 $. Since the normal line is perpendicular to the tangent line, its slope will be the negative reciprocal of the slope of the tangent line. Finally, we use the point-slope form to write the equation of the line.

1. Find the derivative of $ f(x) $:
$$ f'(x) = \frac{d}{dx}\left(\frac{x}{x-2}\right) $$

2. Evaluate the derivative at $ x = 1 $ to find the slope of the tangent line:
$$ f'(1) = \frac{d}{dx}\left(\frac{x}{x-2}\right) \bigg|_{x=1} $$

3. Once we have the slope of the tangent line, we find the slope of the normal line, which is the negative reciprocal of the slope of the tangent line.

4. We then use the point-slope form of the equation of a line to find the equation of the normal line, using the given point $ (1, f(1)) $.

The equation of the line normal to $ f(x) = \frac{x}{x-2} $ at $ x = 1 $ can be found using these steps.