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NDA Straight Lines

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This page covers NDA Straight Lines with complete concept notes, 26 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Straight Lines— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

A straight line is the shortest path between two points. In coordinate geometry, we study lines using equations and formulas. This topic appears in almost every NDA paper with 2-3 questions guaranteed. Core Concept: Every straight line can be written as an equation. The most common forms are: y = mx + c (slope-intercept form), ax + by + c = 0 (general form), and (y - y1) = m(x - x1) (point-slope form).

2.Key Rules

Core rules you must know cold

Slope (m) tells us how steep the line is. Positive slope means line goes upward, negative slope means downward. Parallel lines have equal slopes. Perpendicular lines have slopes that multiply to give -1.

3.Formula Block

Memorise — at least one formula appears in every paper

• Slope between two points: m = (y2 - y1)/(x2 - x1)

• Distance between points: d = √[(x2-x1)² + (y2-y1)²]

• Angle between lines: tan θ = |(m1-m2)/(1+m1m2)|

• Section formula: ((mx2+nx1)/(m+n), (my2+ny1)/(m+n))

• Area of triangle: (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|

4.Exam Patterns

What examiners ask — read before attempting PYQs

Shortcut 1 - Perpendicular Lines

If line has slope m, perpendicular line has slope -1/m. For line ax + by + c = 0, perpendicular line is bx - ay + k = 0

Shortcut 2 - Quick Area

For triangle with vertices at origin and two other points, area = (1/2)|x1y2 - x2y1|.

5.Worked Example

Solve this step-by-step before moving on

1

Step 1

Find slope m = (7-3)/(4-2) = 4/2 = 2

2

Step 2

Use point-slope form with point (2,3)

3

Step 3

y - 3 = 2(x - 2)

4

Step 4

y - 3 = 2x - 4

5

Step 5

y = 2x - 1 Answer: y = 2x - 1 or 2x - y - 1 = 0 Worked Example 2: Find distance between parallel lines 3x + 4y + 5 = 0 and 3x + 4y - 7 = 0.

1

Step 1

Both lines have form 3x + 4y + c = 0, so they are parallel

2

Step 2

Use distance formula: d = |c1 - c2|/√(a² + b²)

3

Step 3

Here c1 = 5, c2 = -7, a = 3, b = 4

4

Step 4

d = |5 - (-7)|/√(3² + 4²) = 12/√25 = 12/5 Answer: 12/5 units Shortcut 3 - Equation from intercepts: If line cuts x-axis at (a,0) and y-axis at (0,b), equation is x/a + y/b = 1.

6.Exam Traps

Common mistakes students make — avoid these

- The #1 trap: Students forget that for perpendicular lines, the product of slopes is -1, NOT the sum. Many write m1 + m2 = -1 instead of m1 × m2 = -1. This single mistake costs marks in every exam. Another frequent error is sign mistakes while applying section formula.

Always check if the ratio is internal (same signs) or external (opposite signs). Remember that when finding perpendicular distance from point to line ax + by + c = 0, the formula is |ax1 + by1 + c|/√(a² + b²). The modulus sign is crucial - distance cannot be negative. For intersection of two lines, solve the equations simultaneously. If you get 0 = 0, lines are identical.

If you get contradiction like 0 = 5, lines are parallel.

Key Points to Remember

Slope of line through (x1,y1) and (x2,y2) is m = (y2-y1)/(x2-x1)
Parallel lines have equal slopes, perpendicular lines have slopes with product = -1
General form of line is ax + by + c = 0, slope-intercept form is y = mx + c
Distance between two points = √[(x2-x1)² + (y2-y1)²]
Perpendicular distance from point (x1,y1) to line ax + by + c = 0 is |ax1 + by1 + c|/√(a² + b²)
Section formula: point dividing line joining (x1,y1) and (x2,y2) in ratio m:n is ((mx2+nx1)/(m+n), (my2+ny1)/(m+n))
Area of triangle with vertices (x1,y1), (x2,y2), (x3,y3) = (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|
Equation from intercepts: if line cuts axes at (a,0) and (0,b), equation is x/a + y/b = 1
Angle between lines with slopes m1 and m2: tan θ = |(m1-m2)/(1+m1m2)|
Distance between parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is |c1-c2|/√(a² + b²)

Exam-Specific Tips

For perpendicular lines, product of slopes equals -1
Slope of x-axis is 0, slope of y-axis is undefined (∞)
Two lines are parallel if their slopes are equal and they don't intersect
Distance formula: d = √[(x2-x1)² + (y2-y1)²]
Midpoint formula: ((x1+x2)/2, (y1+y2)/2)
If triangle area is zero, the three points are collinear
Intercept form of line equation: x/a + y/b = 1
For line passing through origin, equation is y = mx (no constant term)

Practice MCQs

Straight Lines — Practice Questions

26graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 26

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Practice 1easy

The distance between the parallel lines 3x + 4y = 10 and 3x + 4y = 25 is:

Practice 2easy

The angle between the lines y = 2x + 3 and y = -x/2 + 5 is:

Practice 3easy

The foot of the perpendicular from the point (2, 3) to the line x + 2y = 5 is:

Practice 4easy

The point of intersection of the lines x + 2y = 5 and 2x - y = 5 is:

Practice 5easy

The equation of the line passing through the points (1, 2) and (3, 6) is:

Practice 6easy

The angle between the lines 2x + y = 5 and x - 2y = 3 is:

Practice 7easy

A line passes through the point (4, 5) and makes an angle of 45° with the positive x-axis. Its equation is:

Practice 8easy

The equation of the straight line passing through the point (2, 3) and perpendicular to the line 3x + 4y = 7 is:

Practice 9easy

The distance between the parallel lines 2x + 3y = 5 and 2x + 3y = 11 is:

Practice 10easy

The slope of the line passing through the points A(−1, 4) and B(3, −2) is:

Practice 11easy

The equation of the line with slope 2 and y-intercept −3 is:

Practice 12easy

The equation of the straight line passing through the point (2, 3) and perpendicular to the line 2x + 3y = 5 is:

Practice 13medium

The distance between the parallel lines 3x + 4y - 10 = 0 and 3x + 4y + 5 = 0 is:

Practice 14medium

The angle between the lines 2x + y - 4 = 0 and x - 2y + 3 = 0 is:

Practice 15medium

The foot of the perpendicular from the point (3, 4) to the line 2x - y + 1 = 0 is:

Practice 16medium

The equation of the line passing through the point of intersection of the lines 2x + 3y - 5 = 0 and x - 2y + 1 = 0, and perpendicular to the line 3x - 4y + 7 = 0 is: