Find an equation of the tangent line to the astraoid at the (3(3)^1/2, 1)

Question

Find an equation of the tangent line to the astraoid at the (3(3)^1/2, 1)

 

x^2/3 + y^2/3 = 4

Answer 1

This is an interesting functional form, but the standard approach still works.

 

The analytic form of the derivative dy/dx can be found by the method of implicit differentiation.  The result  is

 

  dy/dx = - (x/y)^1/3   .    Evaluating this at  ( 3 √3 , 1)  yields - √3  ; this is the slope of the tangent line.

 

  The last step is the point slope form for the equation of a line   y -y₁ = m ( x - x₁)    Plugging in gives

   y - 1 = - √3 (x - 3√3)      Rearranging gives

 

   y = -√3 x +10