Question

The parametric equation of a line is given by, \[x=-2+\dfrac{t}{\sqrt{10}}\] and \[y=1+\dfrac{3t}{\sqrt{10}}\]. Then, for the line
(a) Intercept on the x – axis = \[\dfrac{7}{3}\]
(b) Intercept on the y – axis = -7
(c) Slope of the line = \[{{\tan }^{-1}}\dfrac{1}{3}\]
(d) Slope of the line = \[{{\tan }^{-1}}3\]

Answer 1

Hint: From the \[{{1}^{st}}\] equation find value of \[\dfrac{t}{\sqrt{10}}\] and substitute in \[{{2}^{nd}}\] equation. You will the equation in the form of slope – intercept form of line y = mx + c. Find the x – intercept by putting y = 0 and y – intercepting by putting, x = 0 and slope m from the equation formed.

Complete step-by-step answer:
We have been given two equations as the parametric equation of a line. The equations are,
 \[x=-2+\dfrac{t}{\sqrt{10}}\] - (1)
 \[y=1+\dfrac{3t}{\sqrt{10}}\] - (2)
From (1) we can say that, \[\dfrac{t}{\sqrt{10}}=x+2\]. Let us put this value in equation (2).
\[\begin{align}
  & y=1+\dfrac{3t}{\sqrt{10}}=1+3\left( x+2 \right) \\
 & \therefore y=1+3x+6 \\
\end{align}\]
\[y=3x+7\] - (3)
Thus we got an equation, which is similar to the slope of the line, y = mx + c.
Let us find the x – intercept, so put y = 0 in equation (3). We get,
\[\Rightarrow 0=3x+7\Rightarrow x=\dfrac{-7}{3}\]
Now let us find the x – intercept, so put x = 0 in the equation (3).
\[\Rightarrow y=3\times 0+7\Rightarrow y=7\]
From the general form and equation (3), we get,
Slope of line = m = 3.
The slope of the line can also be written as \[\tan \theta \].
\[\begin{align}
  & \therefore \tan \theta =3 \\
 & \theta ={{\tan }^{-1}}3 \\
\end{align}\]
Thus we got the intercept on the x – axis = \[\dfrac{-7}{3}\].
The intercept on the y – axis = 7.
Slope of the line = \[{{\tan }^{-1}}3\].
Hence option (d) is the correct answer.

Note: The x – intercept is a point where the line crosses the x – axis. Thus at this point y = 0 and the y – intercept is the point where the line crosses the y – axis. At this point x = 0. So remember these two points while finding the intercepts. Take care of signs while finding the intercepts, as options are similar to our results. We might choose options (a) and (b) also as correct if signs are not taken care of.