Straight Lines

Learning Objectives

• Understand the concept of the slope of a line as the tangent of the angle it makes with the positive x-axis and calculate it using the difference in y-coordinates divided by the difference in x-coordinates.
• Derive and apply various forms of the equation of a line, including point-slope form, slope-intercept form, two-point form, and intercept form.
• Calculate the perpendicular distance from a point to a line using the formula involving the absolute value of $Ax_1 + By_1 + C$ divided by the square root of $A^2 + B^2$.
• Determine conditions for parallelism and perpendicularity of lines based on their slopes, recognizing that parallel lines have equal slopes and perpendicular lines have slopes whose product is -1.
• Compute the angle between two intersecting lines using the formula for the tangent of the angle, given by the absolute difference of the slopes divided by 1 plus the product of the slopes.
• Find the distance between two parallel lines using the absolute difference of their constants divided by the square root of $A^2 + B^2$.
• Verify the collinearity of points by ensuring the slope between any two pairs of points is the same.
• Transform the general equation of a line $Ax + By + C = 0$ into standard forms like slope-intercept or intercept form, and identify special cases.
• Analyze lines cutting the coordinate axes, including equal intercepts, intercept sum/product conditions, and line segments between axes.
• Recall coordinate geometry formulas such as the distance formula, section formula, midpoint formula, and area of triangles/quadrilaterals, using zero area for collinearity.
• Solve problems involving the foot of the perpendicular from a point to a line, altitude of a triangle, right bisector of a segment, and perpendicular from the origin.
• Investigate concurrency and intersection of lines by finding intersection points and checking concurrence of three lines.

Chapter Notes

Slope of a Line

• Definition: The slope of a line is defined as the tangent of the angle it makes with the positive x-axis.
• Formula: For a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Special Cases:
  • Horizontal line: $m = 0$
  • Vertical line: $m$ is undefined

Equation of a Line

• Point-Slope Form: For a line with slope $m$ passing through a point $(x_0, y_0)$: $y - y_0 = m(x - x_0)$
• Slope-Intercept Form: For a line with slope $m$ and y-intercept $c$: $y = mx + c$
• Two-Point Form: For a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
• Intercept Form: For a line making intercepts $a$ and $b$ on the x- and y-axes: $\frac{x}{a} + \frac{y}{b} = 1$

Distance from a Point to a Line

• Formula: The perpendicular distance $d$ from a point $(x_1, y_1)$ to the line $Ax + By + C = 0$ is given by: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

Parallel and Perpendicular Lines

• Parallel Lines: Two lines are parallel if their slopes are equal, i.e., $m_1 = m_2$.
• Perpendicular Lines: Two lines are perpendicular if the product of their slopes is $-1$, i.e., $m_1 \cdot m_2 = -1$.

Angle Between Two Lines

• Formula: The tangent of the angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by: $\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$

Distance Between Parallel Lines

• Formula: For two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$, the distance $d$ between them is: $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$

Collinearity of Points

• Condition: 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 the same.

General Equation of a Line and Reduction to Standard Forms

• General Form: $Ax + By + C = 0$
• Reduction: Convert to slope-intercept form $y = mx + c$ or intercept form $\frac{x}{a} + \frac{y}{b} = 1$ by solving for $y$ or finding intercepts.

Intercepts and Lines Cutting Coordinate Axes

• Intercepts: The x-intercept is the point where the line crosses the x-axis, and the y-intercept is where it crosses the y-axis.
• Equal Intercepts: If a line cuts equal intercepts on the axes, then $a = b$ in the intercept form.

Coordinate Geometry Recall and Area-Based Problems

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Area of Triangle: For vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, the area is: $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$

Foot of Perpendicular and Altitude Problems

• Foot of Perpendicular: The foot of the perpendicular from a point to a line can be found using the distance formula and the equation of the line.

Concurrency and Intersection of Lines

• Intersection: The intersection point of two lines can be found by solving their equations simultaneously.
• Concurrency: Three lines are concurrent if they intersect at a single point.