Question

The parametric equation of a line is given by \[x = - 2 + \dfrac{t}{{\sqrt {10} }}\] and \[y = 1 + 3\dfrac{t}{{\sqrt {10} }}\] . 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: We know that equation of line passing through a point \[({x_1},{y_1})\] and having a slope \[m\] is \[(y - {y_1}) = m(x - {x_1})\] . We also know equation of line, which has the intercepts \[a\] and \[b\] on \[x\] and \[y\] axis respectively is \[\dfrac{x}{a} + \dfrac{y}{b} = 1\] . In data given \[x\] and \[y\] values we simplify this and compare the obtained equation with the point-slope equation or intercept equation, we obtain the required result.

Complete step-by-step answer:
Given, \[x = - 2 + \dfrac{t}{{\sqrt {10} }}\] and \[y = 1 + 3\dfrac{t}{{\sqrt {10} }}\]
Now \[x = - 2 + \dfrac{t}{{\sqrt {10} }}\] can written as
(Adding 2 on the both sides of the equation)
 \[ \Rightarrow x + 2 = \dfrac{t}{{\sqrt {10} }}\] ---- (1)
Now, we have \[y = 1 + 3\dfrac{t}{{\sqrt {10} }}\]
Using equation (1) substituting the value of \[\dfrac{t}{{\sqrt {10} }}\] in \[y\] , we get:
 \[ \Rightarrow y = 1 + 3(x + 2)\]
 \[ \Rightarrow y - 1 = 3(x + 2)\]
If we observe the above equation it is in the form \[(y - {y_1}) = m(x - {x_1})\] . That is, the equation of the line passing through a point \[({x_1},{y_1})\] and having a slope \[m\] .
 \[ \Rightarrow m = 3\] and \[({x_1},{y_1}) = ( - 2,1)\] .
The slope of the line is 3.
We know that the slope of a line is the tangent of its inclination, that is
Slope \[m = \tan \theta \]
Substituting the values of \[b\] .
 \[ \Rightarrow 3 = \tan \theta \]
 \[ \Rightarrow \tan \theta = 3\]
 \[ \Rightarrow \theta = {\tan ^{ - 1}}3\]
That is, slope of the line is \[ = {\tan ^{ - 1}}3\]
So, the correct answer is “Option D”.

Note: Remember the equation of line passing through a point and having slope, also the equation of intercept form. They can also ask us to find respective equations. If we obtained the equation after simplifying in the form \[\dfrac{x}{a} + \dfrac{y}{b} = 1\] then intercept on the x- axis is \[a\] and intercept on the y-axis is \[b\] . Careful while simplifying.