IITB Practice Set: 1

Question

Given four distinct points $A, B, C,$ and $D$ on the directed ray denoting the positive $x$-axis, the cross ratio of
$A,B$ with respect to $C,D$ denoted by $(A,B;C,D)$ is defined to be $\frac{AC}{BC}/\frac{AD}{BD}$. Here, $AC$, for example, represents
the length of the directed line segment and thus is considered negative if C has a smaller $x$-coordinate than $A$.

1. The labeling of points is arbitrary. The number of distinct cross ratios is _____.
2. The labeling of points is fixed, and four ordered points $(A,B;C,D)$ with cross-ratio $r$ on the positive $x$-axis are observed from the location $(0,5)$, and the images of these points are formed on the line $y = x$ at locations $(A', B';C',D')$ in the first quadrant. A colleague pretends that this ray of slope $1$ is the $x$-axis and computes the cross-ratio as $p$. The value of of $r/p$ is ______.