Straight Lines

The One Thing

Straight-line geometry represents a line through its slope, a point, or its intercepts, and uses these representations to determine equations, relative positions, angles, and distances. The appropriate form should be selected according to the information given and verified by checking that known points satisfy the resulting equation.

Definitions and Results

• Coordinate Plane: A plane formed by the $x$-axis and $y$-axis, used to locate points with ordered pairs $(x,y)$.

• Slope or Gradient: The measure of the inclination of a line to the positive direction of the $x$-axis. For two points $(x_1,y_1)$ and $(x_2,y_2)$,

$$m=\frac{y_2-y_1}{x_2-x_1},$$ provided $x_2\ne x_1$.

• Angle of Inclination: The angle $\theta$ measured anticlockwise from the positive $x$-axis to a line. The slope is

$$m=\tan \theta .$$

• Horizontal Line: A line parallel to the $x$-axis. Its slope is $0$, and its equation is

$$y=b,$$ where $b$ is a constant.

• Vertical Line: A line parallel to the $y$-axis. Its slope is undefined, and its equation is

$$x=a,$$ where $a$ is a constant. It cannot be expressed using a finite slope.

• Point-Slope Form: The equation of a line with slope $m$ passing through $(x_1,y_1)$ is

$$y-y_1=m(x-x_1).$$

• Two-Point Form: The equation of a line passing through $(x_1,y_1)$ and $(x_2,y_2)$ is

$$y-y_1=\left(\frac{y_2-y_1}{x_2-x_1}\right)(x-x_1),$$ provided $x_2\ne x_1$. Equivalently, when the denominators are nonzero, $$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}.$$

• Slope-Intercept Form: The equation

$$y=mx+c$$ represents a line with slope $m$ and $y$-intercept $c$. A line through the origin with slope $m$ has equation $$y=mx.$$

• Intercept Form: If a line cuts the $x$-axis at $(a,0)$ and the $y$-axis at $(0,b)$, its equation is

$$\frac{x}{a}+\frac{y}{b}=1,$$ provided $a\ne 0$ and $b\ne 0$.

• Normal Form: The equation

$$x\cos \alpha +y\sin \alpha =p$$ describes a line where $p$ is the perpendicular distance from the origin to the line and $\alpha$ is the angle made by the perpendicular with the positive $x$-axis.

• General or Standard Form: The equation

$$Ax+By+C=0$$ represents a straight line, where $A$, $B$, and $C$ are constants and $A$ and $B$ are not both zero.

• Parallel Lines: Two non-vertical lines are parallel if their slopes are equal:

$$m_1=m_2.$$ For $$A_{1}x+B_{1}y+C_1=0\text{and}{A}_{2}x+B_{2}y+C_2=0,$$ the lines are parallel when $$\frac{A_1}{B_1}=\frac{A_2}{B_2}.$$

• Perpendicular Lines: Two non-vertical lines are perpendicular if

$$m_1{m}_2=-1.$$ In general form, the condition is $$A_1{A}_2+B_1{B}_2=0.$$

• Angle Between Two Lines: For lines with slopes $m_1$ and $m_2$,

$$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|,$$ provided $1+m_1{m}_2\ne 0$.

• Distance of a Point from a Line: The perpendicular distance from $(x_1,y_1)$ to

$$Ax+By+C=0$$ is $$\frac{|A{x}_1+B{y}_1+C|}{\sqrt{A^2+B^2}}.$$ The absolute value is required because distance cannot be negative.

• Distance Between Parallel Lines: For

$$Ax+By+C_1=0\text{and}Ax+By+C_2=0,$$ the distance between the lines is $$\frac{|C_1-C_2|}{\sqrt{A^2+B^2}}.$$

Worked Methods

1. Finding the Slope from Two Points

• Identify the two points $(x_1,y_1)$ and $(x_2,y_2)$.
• Subtract the $y$-coordinates in the order $y_2-y_1$.
• Subtract the $x$-coordinates in the same order $x_2-x_1$.
• Calculate

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

• Confirm that $x_2\ne x_1$. If $x_2=x_1$, the line is vertical and its slope is undefined.

2. Finding an Equation Using Point-Slope Form

• Identify the slope $m$ and a point $(x_1,y_1)$ on the line.
• Substitute them into

$$y-y_1=m(x-x_1).$$

• Expand and simplify if another form is required.
• If the line passes through the origin, use

$$y=mx.$$

3. Finding an Equation Using Two-Point Form

• Identify the points $(x_1,y_1)$ and $(x_2,y_2)$.
• Substitute them into

$$y-y_1=\left(\frac{y_2-y_1}{x_2-x_1}\right)(x-x_1).$$

• Alternatively, use

$$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1},$$ when the denominators are nonzero.

• Simplify the resulting equation and verify that both given points satisfy it.

4. Finding an Equation from the Intercepts

• Identify the $x$-intercept $a$ and $y$-intercept $b$.
• Recognise that the line passes through $(a,0)$ and $(0,b)$.
• Substitute the intercepts into

$$\frac{x}{a}+\frac{y}{b}=1.$$

• Use this form only when $a\ne 0$ and $b\ne 0$.

5. Converting an Equation into General Form

• Expand brackets and simplify.
• Collect all terms on one side.
• Write the result as

$$Ax+By+C=0.$$

• Ensure that $A$ and $B$ are not both zero.

6. Finding a Parallel Line

• Determine the slope of the given line, unless it is vertical.
• Use the same slope for the required parallel line:

$$m_{\text{new}}=m_{\text{given}}.$$

• Substitute the required point or other condition into point-slope form.
• If the given line is in general form, use proportional coefficients of $x$ and $y$.
• For a vertical line, retain the form $x=a$; for a horizontal line, retain the form $y=b$.

7. Finding a Perpendicular Line

• Determine the slope of the given line.
• Use the negative reciprocal:

$$m_{\text{new}}=-\frac{1}{m_{\text{given}}}.$$

• Substitute the required point into point-slope form.
• Alternatively, if the lines are in general form, apply

$$A_1{A}_2+B_1{B}_2=0.$$

• Account separately for horizontal and vertical lines: a line perpendicular to a horizontal line is vertical, and a line perpendicular to a vertical line is horizontal.

8. Finding the Angle Between Two Lines

• Determine the slopes $m_1$ and $m_2$.
• Substitute them into

$$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|.$$

• Ensure that $1+m_1{m}_2\ne 0$.
• If $m_1{m}_2=-1$, the lines are perpendicular.

9. Finding the Distance from a Point to a Line

• Write the line in general form:

$$Ax+By+C=0.$$

• Identify the point $(x_1,y_1)$.
• Substitute the point into the numerator:

$$A{x}_1+B{y}_1+C.$$

• Take the absolute value.
• Divide by

$$\sqrt{A^2+B^2}.$$

• The distance is

$$\frac{|A{x}_1+B{y}_1+C|}{\sqrt{A^2+B^2}}.$$

• For the origin, use $(0,0)$, giving

$$\frac{|C|}{\sqrt{A^2+B^2}}.$$

10. Finding the Distance Between Parallel Lines

• Ensure the equations have identical coefficients of $x$ and $y$:

$$Ax+By+C_1=0,Ax+By+C_2=0.$$

• Subtract the constant terms and take the absolute value:

$$|C_1-C_2|.$$

• Divide by

$$\sqrt{A^2+B^2}.$$

• The distance is

$$\frac{|C_1-C_2|}{\sqrt{A^2+B^2}}.$$

Where It Goes Wrong

• Applying

$$m=\frac{y_2-y_1}{x_2-x_1}$$ when $x_2=x_1$; this condition gives a vertical line with undefined slope and equation $x=a$.

• Forgetting that a horizontal line has slope $0$ and equation $y=b$, rather than treating it as a line with an undefined slope.

• Using the reciprocal instead of the negative reciprocal when constructing a perpendicular line; the required condition is $m_1{m}_2=-1$.

• Applying intercept form

$$\frac{x}{a}+\frac{y}{b}=1$$ when $a$ or $b$ is zero, despite the requirement that both intercepts be nonzero.

• Omitting the absolute value in point-line and parallel-line distance formulas, which can produce a negative value even though distance cannot be negative.

• Failing to put an equation into

$$Ax+By+C=0$$ before applying the distance formula, or failing to ensure that the two parallel-line equations have the same $x$- and $y$-coefficients.

What Gets Asked

• Find the slope or gradient of the line through two given points.
• Determine the angle of inclination using $m=\tan \theta$.
• Identify whether a line is horizontal or vertical and state its equation.
• Find the equation of a line using a point and its slope.
• Find the equation of a line through two points.
• Find the equation of a line from its $x$- and $y$-intercepts.
• Find the equation of a line through the origin with a specified slope.
• Convert an equation into general form $Ax+By+C=0$.
• Determine whether two lines are parallel or perpendicular using slopes or general-form coefficients.
• Find the equation of a line parallel or perpendicular to a given line subject to a point or other condition.
• Calculate the angle between two lines.
• Calculate the perpendicular distance from a point to a line.
• Calculate the distance from the origin to a line.
• Calculate the distance between two parallel lines.
• Verify that a proposed line equation contains specified points.