Expert: Paul Klarreich - 3/10/2011

Question
Thank you for your help on the last question-you explained it extremely well. I have anothr one that is stumping me I was hoping you could help me with also.

Show that the sum of the x- and y- intercepts of any tangent line to the curve (sqrt of x + sgrt of y = sqrt of c), is equal to c.

Answer
Questioner: Emily
Country: United States
Category: Calculus
Private: No
Subject: sum of intercepts of tangent lines
Question: Thank you for your help on the last question-you explained it extremely well. I have anothr one that is stumping me I was hoping you could help me with also.

Show that the sum of the x- and y- intercepts of any tangent line to the curve (sqrt of x + sgrt of y = sqrt of c), is equal to c.
...........................................
First a note: (no, not E-flat, a comment, I mean):

When you see a question beginning with the words "Show that", do you immediately reach for your anti-anxiety medication?  Don't.  Just reword the question as:

"Find the sum of the x- and y- intercepts of any tangent line to the curve (sqrt of x + sgrt of y = sqrt of c).  BTW, the answer in the back of the book is c."

Your equation is:

sx + sy = sc   <<< I really hate to type, so...

I think you use implicit differentiation here:

 1     1
---- + ---- dy/dx = 0
2 sx   2 sy

 1     1
---- + ---- dy/dx = 0
2 sx   2 sy

          sy
dy/dx = - ----
          sx

          sc - sx    <<< that is  sy
dy/dx = - --------
          sx

       sx - sc
dy/dx = --------
          sx

Now, at some point on the graph,  (x0,y0)   <<< remember that??

   s(x0) - s(c)
m = -------------
      s(x0)

     s(y0)
m = - -----
     x(x0)


and we find the equation of the T.L.

y - y0 = m(x - x0)

           s(y0)
y - y0 = - -------(x - x0)
           s(x0)

We can simplify that, but perhaps we don't have to:

X-INTERCEPT:  set y = 0, solve for x.

           s(y0)
 - y0 = - -------(x - x0)
           s(x0)

           s(y0)
   y0 =   -------(x - x0)
           s(x0)

s(x0) y0 = s(y0)(x - x0)

s(x0)s(y0) = x - x0

x = s(x0)s(y0) + x0


.................................
Y-INTERCEPT:  set x = 0, solve for y.

           s(y0)
y - y0 = - -------(  - x0)
           s(x0)

           s(y0)
y - y0 =   -------(   x0)
           s(x0)

          
y - y0 =  s(y0) s(x0)
           
y = y0 +  s(y0) s(x0)

OKAY, Now find the sum of those intercepts:

SUM = x + y = s(x0)s(y0) + x0 + y0 +  s(y0) s(x0)

SUM = 2 s(x0)s(y0) + x0 + y0

AHA! Does that look familiar?  Remember this algebra:

(a + b)^2 = a^2 + b^2 + 2ab

SUM = 2 s(x0)s(y0) + s(x0)^2 + s(y0)^2

SUM = ( s(x0) + s(y0) )^2

BUT WAIT, there's more!

(which I will leave to you -- why should I have all the fun?)