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## Q. 1.5

Find a parametric representation of the line passing through the points P = (2, 5) and Q = (-3, 9).

## Verified Solution

Using (9),

$x=x_{1}+t\left(x_{2}-x_{1}\right) \quad \text { and } \quad y=y_{1}+t\left(y_{2}-y_{1}\right)$

we immediately find that

$x=2-5 t \quad \text { and } \quad y=5+4 t$

Note that the problem asked for a parametric representation of the line, since there are several such representations. For example, reversing the roles of P and Q, we obtain the representation [from the new P=(-3,9) and Q=(2,5)]

$x=-3+5 s \quad \text { and } \quad y=9-4 s$

Although these representations look different, they do represent the same line, as is seen by inserting a few sample values for t and s. For example, if t=1, from (10) we obtain the point (-3,9). This point is obtained from (11) by setting s=0.

To obtain another parametric representation, we first observe that the slope-intercept form of the line passing through the points (2,5) and (-3,9) is given by

$\frac{y-5}{x-2}=\frac{9-5}{-3-2}=-\frac{4}{5^{\prime}} \quad \text { or } \quad y=-\frac{4}{5} x+\frac{33}{5}$

Then setting t=x, we obtain the “slope-intercept” parametrization

$x=t, \quad y=-\frac{4}{5} t+\frac{33}{5}$