Question

In ℝ2, you are given the points A(−14,15) and X(13,−17). Find t such that the point C(−12,t) lies on the line through A and X. answer as a ratio not as a decimal

Answer 1

The equation of the line segment between points A and X is

`sA + (1-s)X, s\in[0,1]`

Then we need that the point C satisfy:

`\left[\begin{array}{c}-12\\t\end{array}\right] =\left[\begin{array}{c}-14s\\15s\end{array}\right]+\left[\begin{array}{c}13(1-s)\\-17(1-s)\end{array}\right]`.

This implies that

`-12=-14s-13s+13\\s=(25)/(27)`

We replace the value of s in the equation we get from the second component

`t=15s+17s-17\\t=32(25)/(27)-17\\t=(341)/(27)`

Then the point
`C=(-12,(341)/(27))` lies on the line through A and X.