Expert: Paul Klarreich - 12/8/2009

Question
prove that the sum of two intercept on the axes of any tangent to the curve x^1/2+y^1/2=a^1/2 is constant

Answer
Questioner: anu
Country: Pakistan
Category: Calculus
Private: No
Subject: plz help
Question: prove that the sum of two intercept on the axes of any tangent to the curve

x^1/2+y^1/2=a^1/2 is constant
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Step 1: Find dy/dx.

Step 2: Let the point of tangency be at x = x0, and find m = y'(x0)

Step 3: write the equation of the line using:

y - y0 = m(x - x0)

Step 4:

Find the y-intercept by setting x = 0.
Find the x-intercept by setting y = 0.

Step 4:  Write their sum and finish the proof.
.....................................


Abbreviations:  A = sqrt(a),  sx = x^1/2 = sqrt(x), s(x0) = sqrt(x0)

........................................

If  x^1/2 + y^1/2 = a^1/2, then

y = (A - sx)^2

Step 1: ---------------

y' = 2(A - sx)(-1/2sx)

y' = (A - sx)(-1/sx), which is indeed negative, because sqrt(x) < sqrt(a)

y' =  (sx - A)/sx, which is indeed negative, because sqrt(x) < sqrt(a)

y' =  1 - A/sqrt(x)

Step 2: ---------------

Now let the point of tangency be at x = x0.

m = 1 - A/sqrt(x0)

Step 3:---------------

y0 = (A - s(x0))^2

Equation is

y - y0 = m(x - x0)

y - (A - s(x0))^2 = (1 - A/s(x0))(x - x0)

Step 4:---------------

Get the y-int:

y - (A - s(x0))^2 = (1 - A/s(x0))(- x0)


y = (A - s(x0))^2 + (1 - A/s(x0))(- x0)

y = (A - s(x0))^2 - x0 + As(x0)

y = A^2 + x0 - 2A s(x0) - x0 + As(x0)

y = A^2  - A s(x0)    <<<<<<<<<<<<<<<<<<<<<<  y-intercept



Get the x-int: (set y = 0)

0 - (A - s(x0))^2 = (1 - A/s(x0))(x - x0)

 - (A^2 + x0 - 2As(x0)) = (1 - A/s(x0))(x - x0)

- A^2 - x0 + 2As(x0) = (1 - A/s(x0))(x - x0)

Mult through by sx0:

- A^2 sx0 - x0 s(x0) + 2A x0 = (s(x0) - A)(x - x0)

- A^2 sx0 - x0 s(x0) + 2A x0 = s(x0)x - Ax - x0 s(x0) + Ax0

- A^2 sx0 + A x0 = s(x0)x - Ax

Factor:

A s(x0)(- A +  s(x0)) = (s(x0) - A)x

Cancel:  

A s(x0) = x    <<<<<<<<<<<<<<<<<<  x-intercept

Step 5: ---------------

FINALLY, add them:

x+y =  A^2  - A s(x0) + A s(x0) = A^2, WHICH IS INDEPENDENT OF x0