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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 $=-9$

Answer
Verified

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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