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 $y=x+\sqrt{13}$.

It's actually quite simple.  An ellipse centered at the origin with semimajor and semiminor axes $a$ and $b$ can be transformed to a circle by a suitable scaling of the horizontal axis by a factor of $b/a$.  Since such a linear transformation preserves tangency, the resulting line has a slope that is also scaled by the same factor; i.e., if $y = x + \beta$ is the tangent line, then the transformed line after the scaling has the form $y' = \frac{a}{b}x' + \beta$, where $(x',y') = (\frac{b}{a}x, \frac{b}{a}y)$.  Now, such a line is tangent to the circle $(x')^2 + (y')^2 = b^2$, so we have by similarity of right triangles $$\frac{a}{b} = \frac{\sqrt{\beta^2 - b^2}}{b},$$ or equivalently, $$\beta^2 = a^2 + b^2,$$ and we are done.
