Question

IIf the parametric equation of a line is given by $x=4+{\displaystyle \frac{t}{\sqrt{2}}}$ 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 ${\tan }^{-1}\left({\displaystyle \frac{1}{2}}\right)$
(c) Intercept made by the line on the x-axis $={\displaystyle \frac{9}{2}}$
(d) Intercept made by the line on the y-axis $=-9$

Answer 1

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

The given equations are,
$x=4+{\displaystyle \frac{t}{\sqrt{2}}}$ 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 $x=4+{\displaystyle \frac{t}{\sqrt{2}}}$ by $2$,
$2x=8+{\displaystyle \frac{2t}{\sqrt{2}}}$
$2x=8+\sqrt{2}t$
From this we can write,
$\sqrt{2}t=2x-8$
Now, we can substitute this in the equation $y=-1+\sqrt{2}t$,
$y=-1+(2x-8)$
$y=2x-9$
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 $y=0$,
$0=2x-9$
$x={\displaystyle \frac{9}{2}}$
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 ${\displaystyle \frac{y}{x}}$. 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.