Interesting association between tangent lines of slope one and ellipses Why is it that a tangent line with slope $1$ to an ellipse centered at the origin will have a transformation of $\pm \sqrt{a^2 +b^2}$ where $a$ and $b$ are the major and minor axis of the ellipse?
For example: The tangent line of slope one to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $y=x+\sqrt{13}$.

This has association with the director circle. This is well known that locus of intersection of perpendicular tangents is $$x^2+y^2=\sqrt{a^2+b^2}$$

Now draw a tangent of slope $m$ in first quadrant of ellipse. Now draw a tangent at lower symmetrical point in fourth quadrant.

As you can see that they are mirror images of each other, mirror being the x-axis. Their slopes are additive inverses, i.e., negative of each other.
You want one slope to be $1$, the other $-1$ you get for no charge!
It is another very well known fact that two lines are perpendicular if $$m_1m_2=-1$$
Clearly, $1$ and $-1$ are possible solutions!
This will look a bit like :

So, that is how you end up with that mysterious factor of $\sqrt{a^2+b^2}$ every time you take $m=1$. The algebraic $x$ and $y$ intercepts come out to be $\pm\sqrt{a^2+b^2}$.
