Show that for all values of m, the straight lines with equations $\displaystyle y=mx\pm \sqrt(b^2+a^2m^2)$ are tangents to the ellipse with equation $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Heelp
The system formed by the equation of the line and the equation of the ellipse has a single solution, therefore the line intersects the ellipse in a single point.
You might find it simpler to multiply on both sides of the equation to get $\displaystyle b^2x^2+ a^2y^2= a^2b^2$ so you can use implicit differentiation: $\displaystyle 2b^2x+ 2a^2yy'= 0$ and $\displaystyle y'= -\frac{b^2x}{a^2y}$.