# Chaitanya's Random Pages

## January 15, 2011

### The parabola as an analogue multiplier

Filed under: mathematics — ckrao @ 2:10 am

Suppose you had the parabola $y = x^2$ at your disposal. Here is a neat way of using it to multiply two positive numbers a and b. I saw this at the Wild About Math blog.

Locate the points $(-a,a^2)$ and $(b,b^2)$ on the parabola $y = x^2$. Then the line joining these two points has y-intercept ab. 🙂

Why is this true? Try to figure it out for yourself before reading on.

One way of seeing it is in observing that the y-intercept will be a weighted sum of the heights $a^2$ and $b^2$. Given the points’ distances from the y axis are a and b, the weights are in the ratio b:a and since they sum to 1, they must be b/(a+b) and a/(a+b). Hence the y-intercept is

$\displaystyle a^2.\frac{b}{a+b} + b^2.\frac{a}{a+b} = \frac {ab(a+b)}{a+b} = ab.$