First slide

Cartesian plane

Question

If the point $(x_{1} + t (x_{2} - x_{1}), y_{1} + t (y_{2} - y_{1})$ divides the join of $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ internally, then

Easy

A

$t < 0$
B

$0 < t < 1$
C

$t > 1$
D

$t = 1$

Solution

The co-ordinates of the point of division are
$\begin{array}{l} (x_{1} + t (x_{2} - x_{1}), y_{1} + t (y_{2} - y_{1})) \\ = (\frac{(1 - t) x_{1} + t x_{2}}{(1 - t) + t}, \frac{(1 - t) y_{1} + t y_{2}}{(1 - t) + t}) \end{array}$
Clearly, these are the coordinates of the point dividing the jo of $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ in the ratio $t : (1 - t)$ . For the internal division, we must have
$t > 0$ and $1 - t > 0 \Rightarrow 0 < t < 1$

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