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

Slope of a line

First, talk in intuitive terms about what is meant by slope. Give real-life examples of slope such as the slope of
the roof of a house, a road going up a hill, or a ladder leaning against a building. Explain that we can assign a
number that allows us to measure the steepness of a straight line. Also, say that the greater the absolute
value of this number, the steeper the line will be.

Slope of a non-vertical line L is the tangent of the angle θ, which the line L makes with the positive direction of
x-axis. In particular,

(a) Slope of a line parallel of x-axis is zero.

(b) Slope of a line parallel to y-axis is not defined.

(c) Slope of a line equally inclined to the axis is −1 or 1.

(d) Slope of a line making equal intercepts on the axis is −1.

Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC
(g) Slopes of two parallel (non-vertical) lines are equal. If m1, m2 are the slopes, then m1 = m2.

(h) If m1 and m2 be the slopes of two perpendicular lines (which are oblique), then m1m2 = - 1.


Straight line

Straight-line equations, or "linear" equations, graph as straight lines, and have simple variables with no
exponents on them. If you see an equation with x and y, then you're dealing with a straight-line equation.

An equation of the form ax + by + c = 0 is called the general equation of a straight line, where x and y are
variable and a, b, c are constants.

Equation of a line parallel to X axis or Y - axis

(i) Equation of any line parallel to x-axis is y = b, b being the directed distance
of the line from the x-axis. In particular equation of x-axis is y = 0

(ii) Equation of any line parallel to y-axis is x = a, a being the directed distance
of the line from the y-axis. In particular equation of y-axis is x = 0.

(a) One point form

Equation of a line (non-vertical) through the point (x1, y1) and having
slope m is
y - y1 = m (x - x1).

 

(b) Two-point form

Equation of a line (non-vertical) through the points (x1, y1) and (x2, y2) is
Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC

(c) Slope-intercept form

Equation of a line (non-vertical) with slope m and cutting off an intercept c from the y-axis is
y = m x + c.


(d) Intercept form

Equation of a line (non-vertical) with slope m and cutting off intercepts a and b from the x-axis
and y-axis respectively is Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC

 

Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC


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. As line intersects x-axis at A (10, 0)
⇒ length of intercept on x-axis, a = 10
Similarly length of intercept on y-axis, b = 10
∴ Using intercept form, equation of line is Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC
or 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. The slope of the line through (- 2, 3) and (- 5, - 6) is m = Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC = 3


⇒ The slope m1 of the required line = Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC

By point - slope form, Y + 3 = Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC
⇒ 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) Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC


Sol. The line passes through the points (- 3, 7) and (0, - 2).
∴ Slope of the line = Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC  = - 3. Answer: (3)


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

Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC

• Distance between parallel lines ax + by + c = 0 and ax + by + d = 0
Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC

• The angle between two lines y = m1x + b1 and y = m2x + b2 is given by
Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC

• The equation a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 represent the same line if
Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC


Concurrent Lines:

Three or more lines are said to be concurrent lines when all of them pass through a common point.


 

The document Examples: Slope of a Line & Straight Line | CSAT Preparation - UPSC is a part of the UPSC Course CSAT Preparation.
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FAQs on Examples: Slope of a Line & Straight Line - CSAT Preparation - UPSC

1. What is the formula to calculate the slope of a line?
Ans. The formula to calculate the slope of a line is given by: slope = (change in y-coordinates)/(change in x-coordinates). It can also be represented as slope = (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
2. How do you determine if a line is straight?
Ans. A line is considered straight if all the points on the line lie in a straight path with no curvature. If the slope of a line remains constant throughout, it indicates a straight line. In other words, if the line does not bend or deviate, it can be classified as a straight line.
3. Can a line have a slope of zero?
Ans. Yes, a line can have a slope of zero. A slope of zero indicates that the line is horizontal, meaning it is parallel to the x-axis. In this case, the line does not rise or fall, and all the points on the line have the same y-coordinate.
4. What does a negative slope represent in a line?
Ans. A negative slope in a line indicates that the line is decreasing as it moves from left to right. It means that as the x-coordinate increases, the y-coordinate decreases. In graphical terms, a line with a negative slope slopes downwards from left to right.
5. How can the slope of a line be used in real-life situations?
Ans. The slope of a line has various real-life applications. For example, in physics, it can determine the speed or velocity of an object. In economics, it can represent the rate of change in demand or supply. In engineering, slope calculations are used in designing ramps, roads, and structures. Additionally, the slope can represent the steepness of a hill or a road.
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