Straight Lines: Equations, Slopes, Angles, Distance, Collinearity, Parallelism

TAKEAWAYS:

• Understand the slope of a line and how it relates to its angle of inclination.
• Learn the four standard forms of the equation of a straight line: slope-intercept, point-slope, two-point, and intercept forms.
• Master the distance formula between two points and the distance from a point to a line.
• Know how to determine if three points are collinear or if two lines are parallel or perpendicular.
• Apply these concepts to solve real-world problems involving geometry and coordinate geometry.

---

What is a Straight Line?

A straight line is the shortest path between two points. In coordinate geometry, it can be represented by an equation involving x and y. The general equation of a straight line is:

$$Ax+By+C=0$$

where $A$, $B$, and $C$ are constants, and $A$ and $B$ are not both zero.

Slope of a Straight Line

The slope (often denoted by $m$) of a line measures its steepness. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

If two points on a line are $(x_1,y_1)$ and $(x_2,y_2)$, then the slope $m$ is:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

Example 1: Find the slope of the line passing through the points $(-3,4)$ and $(2,-1)$.

Solution: $$m=\frac{y_2-y_1}{x_2-x_1}=\frac{-1-4}{2-(-3)}=\frac{-5}{5}=-1$$

Visualization:

Angle of Inclination

The angle of inclination ($\theta$) of a line is the angle it makes with the positive direction of the x-axis. The slope $m$ of the line is related to $\theta$ by:

$$m=\tan \theta$$

Example 2: Find the angle of inclination of a line whose slope is $m=\sqrt{3}$.

Solution: $$\tan \theta =\sqrt{3}⟹\theta =60^{\circ }$$

---

Equations of Straight Lines

There are four standard forms to represent the equation of a straight line:

1. Slope-Intercept Form

The slope-intercept form is: $$y=mx+c$$ where:

• $m$ is the slope of the line,
• $c$ is the y-intercept (the point where the line crosses the y-axis).

Example 3: Write the equation of a line with slope $2$ and y-intercept $-3$.

Solution: $$y=2x-3$$

2. Point-Slope Form

The point-slope form is: $$y-y_1=m(x-x_1)$$ where $(x_1,y_1)$ is a point on the line and $m$ is the slope.

Example 4: Find the equation of the line passing through $(4,-1)$ with slope $3$.

Solution: $$y-(-1)=3(x-4)⟹y+1=3x-12⟹y=3x-13$$

3. Two-Point Form

The two-point form is: $$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$$ where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.

Example 5: Find the equation of the line passing through $(-2,3)$ and $(1,-4)$.

Solution: $$\frac{y-3}{-4-3}=\frac{x-(-2)}{1-(-2)}⟹\frac{y-3}{-7}=\frac{x+2}{3}$$ Cross-multiplying: $$3(y-3)=-7(x+2)⟹3y-9=-7x-14⟹7x+3y+5=0$$

4. Intercept Form

The intercept form is: $$\frac{x}{a}+\frac{y}{b}=1$$ where:

• $a$ is the x-intercept (the point where the line crosses the x-axis),
• $b$ is the y-intercept.

Example 6: Find the equation of the line with x-intercept $5$ and y-intercept $-2$.

Solution: $$\frac{x}{5}+\frac{y}{-2}=1⟹\frac{x}{5}-\frac{y}{2}=1$$ Multiplying through by 10: $$2x-5y=10$$

---

Distance Between Two Points

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$

Example 7: Find the distance between $(-1,2)$ and $(3,-4)$.

Solution: $$d=\sqrt{(3-(-1){)}^2+(-4-2{)}^2}=\sqrt{(4{)}^2+(-6{)}^2}=\sqrt{16+36}=\sqrt{52}=2\sqrt{13}$$

---

Distance from a Point to a Line

The distance $D$ from a point $(x_0,y_0)$ to a line $Ax+By+C=0$ is:

$$D=\frac{|A{x}_0+B{y}_0+C|}{\sqrt{A^2+B^2}}$$

Example 8: Find the distance from the point $(2,3)$ to the line $3x-4y+12=0$.

Solution: $$D=\frac{|3(2)-4(3)+12|}{\sqrt{3^2+(-4{)}^2}}=\frac{|6-12+12|}{\sqrt{9+16}}=\frac{|6|}{5}=\frac{6}{5}$$

---

Collinearity of Points

Three points $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ are collinear if the slope between any two pairs of points is equal. Alternatively, the area of the triangle formed by the three points is zero:

$$\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|=0$$

Example 9: Check if the points $(1,2)$, $(3,4)$, and $(5,6)$ are collinear.

Solution: Calculate the slope between $(1,2)$ and $(3,4)$: $$m_1=\frac{4-2}{3-1}=1$$ Calculate the slope between $(3,4)$ and $(5,6)$: $$m_2=\frac{6-4}{5-3}=1$$ Since $m_1=m_2$, the points are collinear.

---

Parallel and Perpendicular Lines

Two lines are parallel if their slopes are equal ($m_1=m_2$). Two lines are perpendicular if the product of their slopes is $-1$ ($m_1\cdot m_2=-1$).

Example 10: Determine if the lines $2x+3y=6$ and $4x+6y=12$ are parallel.

Solution: Rewrite both equations in slope-intercept form:

1. $2x+3y=6⟹3y=-2x+6⟹y=-\frac{2}{3}x+2$
2. $4x+6y=12⟹6y=-4x+12⟹y=-\frac{2}{3}x+2$

Both lines have the same slope ($m=-\frac{2}{3}$), so they are parallel.

---

Summary Table of Key Formulas

---

NEB-Style Questions for Practice

1. Find the slope of the line passing through the points $(5,-2)$ and $(-3,4)$.
2. Write the equation of the line with slope $-2$ and passing through the point $(1,3)$.
3. Find the equation of the line passing through the points $(2,-1)$ and $(-4,5)$.
4. Calculate the distance between the points $(-1,3)$ and $(4,-2)$.
5. Find the distance from the point $(0,0)$ to the line $3x-4y+12=0$.
6. Check if the points $(1,1)$, $(2,3)$, and $(3,5)$ are collinear.
7. Determine if the lines $x+2y=4$ and $2x+4y=8$ are parallel or perpendicular.
8. Find the angle of inclination of the line $y=\sqrt{3}x+2$.

---

Exam Tip

• Memorize the formulas for slope, equations of lines, distance, and collinearity. These are frequently tested in NEB exams.
• Practice converting between different forms of line equations (e.g., slope-intercept to standard form).
• Draw diagrams for problems involving lines, slopes, and distances. Visualizing the problem helps in understanding and solving it correctly.
• Check your calculations carefully, especially when dealing with slopes and distances, as small arithmetic errors can lead to incorrect answers.
• Understand the geometric interpretation of parallel and perpendicular lines. This will help in solving problems involving angles between lines.

---