perplexus.info :: Just Math : Smaller product

Smaller product (Posted on 2019-01-29)

If
x+1/y=12 ..... (i)
y+1/x=3/8 ... (ii)

What is the smallest value of x*y?

No Solution Yet Submitted by Ady TZIDON

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Solution

| Comment 1 of 2

I couldn't find a clever way so this is just straightforward.

We have a system of two conic equations. Find the solutions and pick the one that minimizes xy.

Solve both equations for y and substitute to get 1/(12-x)=3/8-1/x which simplifies to the quadratic 0=x^2-12x+32=(x-4)(x-8)

The solutions to the system (x,y) and their product:

(4,1/8) and xy=1/2

(8,1/4) and xy=2

So the smallest xy is 1/2.

Posted by Jer on 2019-01-29 08:53:08

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