Expert: Paul Klarreich - 3/25/2008

Question
Consider the graph defined implicitly by x^2 + 3xy + y^2 = 5. A) Find the coordinates of any point on this graph where the slope of a tangent line is -1. B)Are there any points on this graph such that a tangent line is horizontal? Justify your answers.

Answer
Questioner:   Connie
Category:  Calculus
Private:  No
 
Subject:  implicit differentiation
Question:  Consider the graph defined implicitly by x^2 + 3xy + y^2 = 5. A) Find the coordinates of any point on this graph where the slope of a tangent line is -1. B)Are there any points on this graph such that a tangent line is horizontal? Justify your answers.
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Hi, Connie,

My instructions to questioners include:

(5) Tell me what you already did, and what you are studying --


Since you didn't, I conclude that you just don't know how to get started, so I'll confine myself to that.

Differentiate implicitly:

x^2 + 3xy + y^2 = 5.  

2x + 2x dy/dx + 3y + 2y dy/dx = 0

Solve for dy/dx:

+ 2x dy/dx + 2y dy/dx = - 2x - 3y
        - 2x - 3y
dy/dx = -----------
         2x + 2y

Now set that equal to:

-1 for part A.

After some algebra, you will get  y = 0 and x = anything.  Combine that with the first equation and you will get  x = +-sqrt(5).


0 for part B.

After some algebra, you will get  y = -2/3x .  Combine that with the first equation.  Show that the resulting equation has no solutions.