Let and be positive reals.
Consider all pairs of positive reals subject to
What is the minimum value of
Some of you might recognize this problem, with the specific case of .
The purpose of this note is to unravel this question, working from what has been given to try and find a logical path.
Observation 1: Recognize the final expression of as the perimeter of a right angled triangle with legs and . Let's consider a right triangle with vertices .
Observation 2: The condition that implies that the point lies on the line . This is the "intercept-intercept" equation of a line, which is often not taught explicitly.
Hence, we can rephrase the question as follows:
Consider all right triangles with legs on the x and y axis, and whose hypotenuse passes through the point . Which of these triangle has the smallest perimeter?
Consider a circle that lies that passes through the point , and is tangential to the x-axis and the y-axis. Specifically, if the circle has radius , it has center , and length gives us
This has solutions . The solution corresponding to a minus sign would correspond to a circle whose center is "between" and . The solution corresponding to the plus sign would correspond to the circle "on the other side". Let the larger circle be (see diagram below).
Claim: The triangle with minimal perimeter is tangential to at .
Let's study this triangle. In the diagram, this triangle is denoted by (blue line). Observe that
Proof of claim: Take any other line (yellow line above) through , which cuts the axis at . Then, it is not tangential to , hence cuts again.
Take a parallel line to , which is tangential to at , which is contained beneath . This line cuts the axis at . Then, perimeter of is greater than perimeter of (by scaling), which is equal to (similar argument as perimeter of ), which is equal to perimeter of .
Hence, the minimal perimeter occurs in triangle .
In conclusion, the minimal value is . In the case where , we get .