Find the third proportional to xy+yx and x2+y2 - Mathematics

Find the third proportional to `x/y + y/x` and `sqrt(x^2 + y^2)`

Sum

Let the required third proportional be p.

`=> x/y + y/x, sqrt(x^2 + y^2)`, p are in continued proportion

`=> x/y + y/x : sqrt(x^2 + y^2) = sqrt(x^2 + y^2) : p`

`=> p(x/y + y/x) = (sqrt(x^2 + y^2))^2`

`=> p((x^2 + y^2)/(xy)) = x^2 + y^2`

`=>` p = xy

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