- Thread starter
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#### sweatingbear

##### Member

- May 3, 2013

- 91

**Problem:**

The sum of two real numbers is \(\displaystyle 1\). What is the minimum value of the sum of the squares of the two numbers?

I have already managed to solve the problem algebraically (by substitution and completing-the-square we arrive at a minimum value of \(\displaystyle 0.5\)), but what I am interested in is a graphical approach.

We have \(\displaystyle x +y = 1\) and wish to find the minimum of \(\displaystyle x^2 + y^2\), which we can call \(\displaystyle f(x,y)\). So we have

\(\displaystyle \begin{cases}

x +y = 1 \\

x^2 + y^2 = f(x,y) \, .

\end{cases}\)

In a (cartesian) coordinate system these two equations represent a line and a circle respectively. Therefore the problem boils down to figuring out what the minimum radius of the circle is \(\displaystyle x^2 + y^2 = f(x,y)\), but this is where I am unable to continue. How can one graphically find the minimum radius of the circle?