Examples Slope of a Line & Straight Line - CSAT Preparation - UPSC PDF

Slope of a line

The slope of a straight line measures its steepness and direction. Intuitively, the slope tells us how much the vertical coordinate (y) changes for a given change in the horizontal coordinate (x). Real-life examples include the slope of a roof, the incline of a road up a hill, or a ladder leaning against a wall. A larger absolute value of the slope corresponds to a steeper line.

For a non-vertical line L that makes an angle θ with the positive x-axis, the slope is the tangent of that angle:

m = tan θ

• Horizontal line: The slope of any line parallel to the x-axis is 0.
• Vertical line: The slope of any line parallel to the y-axis is not defined.
• Line equally inclined to both axes: The slope is 1 or -1.
• Line making equal intercepts on axes (opposite signs): The slope is -1.
• Parallel lines: Two non-vertical lines are parallel if and only if their slopes are equal. If the slopes are m₁ and m₂, then m₁ = m₂.
• Perpendicular lines (oblique): If two oblique lines have slopes m₁ and m₂, they are perpendicular if and only if m₁ m₂ = -1.

Straight line

A straight line in the plane is the graph of a linear equation in x and y. A general form for the equation of a straight line is

ax + by + c = 0, where x and y are variables and a, b, c are constants, not all a and b zero.

Equation of a line parallel to the axes

• Parallel to x-axis: Any line parallel to the x-axis has equation y = b, where b is the (directed) distance from the x-axis. In particular the x-axis itself is y = 0.
• Parallel to y-axis: Any line parallel to the y-axis has equation x = a, where a is the (directed) distance from the y-axis. In particular the y-axis itself is x = 0.

Forms of the equation of a line

One-point (point-slope) form

Equation of a non-vertical line through the point (x₁, y₁) with slope m is

y - y₁ = m (x - x₁).

Two-point form

Equation of the line passing through two distinct points (x₁, y₁) and (x₂, y₂) is given by the two-point form:

Slope-intercept form

Equation of a line with slope m and y-intercept c is

y = m x + c.

Intercept form

Equation of a line that cuts the x-axis at a and the y-axis at b (a ≠ 0, b ≠ 0) is

Examples

Ex.1 Line intersects x axis at A (10, 0) and y-axis at B (0, 10). Find the equation of the line.
(1) x + y = 10 
(2) x + y = 20 
(3) x = - y 
(4) None of these

Sol.

The line intercepts the x-axis at A(10, 0) so the x-intercept a = 10.

The line intercepts the y-axis at B(0, 10) so the y-intercept b = 10.

Using the intercept form of the line, the equation is:

This simplifies to x + y = 10.

Answer: (1)

Ex.2 Find the equation of the straight line passing through the point (- 2, - 3) and perpendicular to
 the line through (- 2, 3) and (- 5, - 6).
(1) X + 2 Y + 8 = 0 
(2) X + 3Y + 11 = 0 
(3) X - 3Y = 7 
(4) X + 3Y = 11

Sol.

Compute the slope of the line passing through (-2, 3) and (-5, -6).

The slope of that line is 3.

Therefore the slope m₁ of the required line (perpendicular to the above) satisfies

So m₁ = -1/3.

Use the point-slope form with the required line passing through (-2, -3):

Simplify to the standard form:

X + 3Y + 11 = 0.

Answer: (2)

Ex.3 Find the slope of the line passing through (- 3, 7) having Y-intercept - 2.
(1) - 5 
(2) 2 
(3) - 3 
(4)  

Sol.

The line passes through the points (-3, 7) and (0, -2).

The slope is -3.

Answer: (3)

Some important results

• Length of perpendicular from the point (x₁, y₁) to the line ax + by + c = 0 is

• Distance between two parallel lines ax + by + c = 0 and ax + by + d = 0 is

• Angle between two lines y = m₁x + b₁ and y = m₂x + b₂ is given by

• Condition for two linear equations to represent the same line: The equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 represent the same line if

Concurrent lines

Three or more lines are concurrent if they all pass through a common point. In analytical geometry, to check concurrency of three lines given by linear equations, one usually solves the system pairwise and verifies that all three meet at the same coordinate point.