# Month: September 2021

## Why Does a² + b² = c² in a Hyperbola?

(A new question of the week) In an ellipse, $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ with focal distance c, parameters a, b, and c all make natural sense, and it is easy enough to see why $$a^2 = b^2 + c^2$$. But in the hyperbola, $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$, the equivalent relationship, $$a^2 + b^2 = c^2$$, is not nearly as natural, nor …

## Trigonometry, Radicals, and a Very Common Error

(A new question of the week) While I was looking through recent questions to choose one to post, I ran across one that deals with an error we see very commonly – in fact, a student I had worked with that very afternoon in face-to-face tutoring had done the same sort of thing. The context …

## What is a Set … Really?

Sometimes the more basic an idea is, the harder it is to define it. It is also very hard to understand a definition in English when you are not a native speaker! We have had some interesting discussions of such issues recently with a student who asks very basic and yet very challenging questions of …

## Trying to Solve a Strange Log Equation

(A new question of the week) We’ll look at a very complicated logarithmic equation, which leads to quartic equations and some very interesting graphs. We won’t find a fully satisfying solution method, but we’ll have some fun trying – and reveal the fallibility of at least one Math Doctor!