Question

# IIf the parametric equation of a line is given by $x=4+\dfrac{t}{\sqrt{2}}$ and $y=-1+\sqrt{2}t$ where $t$ is the parameter, then(a) Slope of the line is ${{\tan }^{-1}}\left( 2 \right)$(b) Slope of the line is ${{\tan }^{-1}}\left( \dfrac{1}{2} \right)$(c) Intercept made by the line on the x-axis $=\dfrac{9}{2}$(d) Intercept made by the line on the y-axis $=-9$

Hint: Simplify the given line equation and substitute it into the parametric equation.

The given equations are,
$x=4+\dfrac{t}{\sqrt{2}}$ and $y=-1+\sqrt{2}t$
We have to rearrange these such that we can formulate an equation in $x$ and $y$ terms. To
change the parametric form of the equation, multiply the equation $x=4+\dfrac{t}{\sqrt{2}}$ by $2$,
$2x=8+\dfrac{2t}{\sqrt{2}}$
$2x=8+\sqrt{2}t$
From this we can write,
$\sqrt{2}t=2x-8$
Now, we can substitute this in the equation $y=-1+\sqrt{2}t$,
$y=-1+\left( 2x-8 \right)$
$y=2x-9$
The options indicate that we need to compute the slope and the intercepts of the line $y=2x-9$. It is
in the form of $y=mx+c$, where $m$ is the slope and $c$ is the y-intercept.
On comparing the equation $y=2x-9$ with the general form, we get the slope as $2$ and the y-
intercept as $-9$.
The x-intercept can be computed by taking $y=0$,
$0=2x-9$
$x=\dfrac{9}{2}$
Looking at the options, we get that both option (c) and (d) are true.

Note: The slope of a line is given by $\tan \theta$ or by $\dfrac{y}{x}$. We get the slope as $2$ for
the line in the question. It means that $\tan \theta =2$ is the slope of the line. The angle of
inclination of a line is represented by ${{\tan }^{-1}}\theta$. So, the options (a) and (b) do not
represent the slope but the angle of inclination of the line.