First slide

Cartesian plane

Question

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

Moderate

A

$t < 0$
B

$0 < t < 1$
C

$t > 1$
D

$t = 1$

Solution

$x c o o r d i n a t e o f t h e p o i n t (x_{1} + t (x_{2} - x_{1}), y_{1} + t (y_{2} - y_{1})) c a n b e r e w r i t e a s$
$\frac{x_{1} + t x_{2} - t x_{1}}{1} = \frac{x_{1} (1 - t) + t x_{2}}{t + (1 - t)}$
$t h i s g i v e s t h e x - c o o r d i a n t e o f a p o i n t w h i c h d i v i d e s t h e l i n e s e g m e n t j o i n i n g t h e g i v e n p o i n t s i n t h e r a t i o t : 1 - t i n t e r n a l l y$
$h e r e 1 - t > 0$
$\Rightarrow$ $t < 1$

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