Parameterize the unit circle by the height of a given chord

Question

The following is an exercise from Theodore Shifrin's differential geometry text:

Parameterize the unit circle (less the point $(-1,0)$) by the length $t$ indicated in figure 1.11.

I feel like this exercise should be pretty trivial but for whatever reason I am struggling with it.

Attempt: Let $\phi$ be the angle between the $x$-axis and the chord formed between $(-1,0)$ and $(x,y)$. Then $\tan\phi=t=\frac{y}{1+x}$, which gives a relationship between $t$, $y$, and $1+x$. However, this is not enough to parameterize the circle yet.

Any hints? I don't want a full solution (like I said, this is probably trivial and I am missing something obvious), just something to point me in the right direction.

Answer 1

Hint: Note that since we are considering a unit circle, $x^2+y^2=1$, so we have the relation $y=\pm\sqrt{1-x^2}$.

Answer 2

Similar triangles AOE, ACB and ABD,

$$\frac{AO}{AE}=\frac{AC}{AB}=\frac{AB}{AD}$$

or

$$\frac{1}{\sqrt{1+t^2}}=\frac{1+x}{AB}=\frac{AB}{2}$$

$$x(t)=\frac{1-t^2}{1+t^2}$$