Chapter 10.1, Problem 25ES

(a) Suppose that the line segment from the point $P\left(x_0,y_0\right)\text{ to }Q\left(x_1,y_1\right)$ is represented parametrically by $\begin{array}{cc}\begin{array}{l}x=x_0+\left(x_1-x_0\right)t,\\ y=y_0+\left(y_1-y_0\right)t\end{array}& \left(0\le t\le 1\right)\end{array}$ and that $R\left(x,y\right)$ is the point on the line segment corresponding to a specified value of t (see the accompanying figure). Show that $t=r/q,$ where r is the distance from P to R and q is the distance from P to Q.

(b) What value of t produces the midpoint between points P and Q?

(c) What value of t produces the point that is three-fourths of the way from P to Q?