Straight Lines - Practice Questions & MCQ

Straight Line

It is a curve such that all points on the line segment joining any two points on it lies on it.
Every equation of first degree in $x$, $y$ represents a straight line. General equation of a straight line is given as $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ where $\mathrm{a}, \mathrm{b}$ and c are real numbers and at least one of a and b is non-zero.

Slope of a Line

Slope of a line tells us the direction in which a line is drawn.

A line in a coordinate plane forms two angles with the x-axis, which are supplementary.

The angle $\theta$ made by the line 'I' with the positive direction of the $x$-axis and measured anticlockwise is called the inclination of the line. $\theta$ lies in the range $\left[0^{\circ}, 180^{\circ}\right)$

If $\theta$ is the angle at which a straight line is inclined to a positive direction of the x -axis, then the slope (or gradient) of this line is defined by $\mathrm{m}=\tan \theta$

Slope will be positive when $\theta$ is an acute angle and it will be negative when $\theta$ is an obtuse angle.

Note

Slope is not defined for a vertical line, as $\theta=90^{\circ}$, and $\tan \left(90^{\circ}\right)$ is not defined.
If a line is equally inclined with the coordinate axes then it will make an angle of $45^{\circ}$ and $135^{\circ}$ wrt positive direction of x -axis. In this case the slope will be $\tan \left(45^{\circ}\right)$ or $\tan \left(135^{\circ}\right)$. i.e. $m=1$ or $m=-1$.

The slope of the line joining two given Points

\begin{equation}
\begin{aligned}
&{ }_{\text {If }} \mathrm{A}\left(\mathrm{x}_1, \mathrm{y}_1\right) \text { and } \mathrm{B}\left(\mathrm{x}_2, \mathrm{y}_2\right) \text { are two points on a straight line then the slope of the line is }\\
&\tan \theta=\frac{B C}{A C}=\frac{y_2-y_1}{x_2-x_1}
\end{aligned}
\end{equation}

Intercepts of a Line

X-Intercept: The x- coordinate of point where the straight line cuts x-axis.

Here straight line cuts X-axis at (x, 0) so

x-intercept is x  (it can be positive or negative)

Length of x-intercept is |x|.

y-intercept: The y- coordinate of point where the straight line cuts y-axis.

Here straight line cuts Y-axis at  (0, y) so

y-intercept is y  (it can be positive or negative)

Length of y-intercept is |y|.