Solution to Problem 9.8

© 2002 by Karl Hahn 		KCT logo (click for Home Page)

The problem is: Find the point (or points) closest to the origin on the conic, 

   x2 + xy  =  C
where C can be any constant.

Step 1: Use the constraint to eliminate one of the variables. It is easier to solve the constraint equation for y in terms of x than vice versa, so that's what we do. From the conic equation we get: 

         C
   y  =     -  x
         x

Step 2: Establish the function for the parameter to be optimized. That would be the distance a point,  (x,y),  is from the origin. We'll call that parameter s. The Pythagorean formula says that 

   s2  =  x2 + y2
You could take the square root of both sides of this to get s all alone on the left. Not only is that not necessary, but the problem is easier to solve if you don't bother doing that (if you did solve it that way, it's ok because that is another correct way to do this one).

Step 3: Eliminate one of the variables using the constraint equation. In the first step you established y as a function of x from the constraint equation. Now we eliminate y by substituting y with the expression you developed for it: 


   s2  =  x2  +



  


C
   -  x
x





2

Step 4: Take the derivative of both sides. Because we have  s2  on the left you will have to use implicit differentiation to take the derivative of that side. You also have to use the chain rule to take the derivative of the parentheses-expression. 

     ds
  2s     =  2x  +  2
     dx





C
   -  x
x









   C
-     -  1
  x2

Step 5: Set the derivative of distance equal to zero. The expression on the left is a product of distance, s, and the derivative of distance with respect to x,  ds/dx.  In order to minimize distance, we need to solve for when the derivative of distance is zero. So we set  ds/dx  equal to zero. That makes the product on the left zero also. Not only that, the product on the left can only be zero if either  ds/dx = 0,  or  s = 0.  But if s were zero, the distance would already be minimized, wouldn't it. So we are completely safe in solving 


  0  =  2x  +  2






C
   -  x
x









   C
-     -  1
  x2
When you find the x that solves this, you have cracked this problem wide open.

Step 6: Use algebra to solve the optimization equation. You can immediately cancel the factor of 2. Then multiply out the two expressions in parentheses. 


   0  =  x  -


  C2
    
  x3

     C     C
  -     +     +  x
     x     x
Taking the cancellations, gathering like terms, and multiplying through by x3 you get 

   0  =  2x4  -  C2



   x4  =


  C2
    
   2


           _
   x  =  ±√C (2-1/4)

Step 7: Back-substitute to find the other variable. Use the original constraint equation to find y from x. I'll let you do the algebra for yourself. When you finish simplifying, you get 

           _
   y  =  ±√C (21/4 - 2-1/4)

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