Answer
457.5k+ views
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.
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.
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