How should I sketch this, for the cases $k = 0$ and $k=1$?

Question

I was trying to solving the problem that involves sketching the tangent line to the ellipse: $\frac{x^2}{a^2} + \frac{b^2}{a^2}=1$.
I was asked by the problem to prove that:
The tangent line to the ellipse to the point $(acos(\alpha), bsin(\alpha))$ can be written in the form:
$y=\frac{-(1-k^2)b}{2ka} x + \frac{b(1+k^2)}{2k}$ where $k=tan(\frac{\alpha}{2})$ . And after that, the problem requires me to test whether when $k=0$ and $k=1$, the line is still a tangent to the ellipse by means of sketch.

May I know how can I sketch the tangent as when $k=0$, the gradient cannot be calculated, since the denominator is $0$. So How can I do this?

Thank you very much for your reply

Answer 1

When $k = 0$, $\alpha = 0$. When $\alpha = 0$, the point of the ellipse is the rightmost point, at which the slope of the tangent is undefined (the tangent line is vertical). So you know a point on and the direction of the tangent line when $k = 0$, so you can sketch that line.

Answer 2

Your expression for the tangent line is derived with a division by $k$, which is not allowe when $k=0$. If you go back to the definition of $k$ you find that $\alpha=0$, so the point in question is $(a,0)$ The tangent is then vertical, so the equation is $x=a$. You can declare that the given equation for the tangent line is wrong because it divides by $k$. You should be able to derive the correct equation if you go back through the definition to before the division by $k$ and set $k=0$