Schooling · Class 11 · Mathematics · Straight Lines · Chapter on a page
CBSE Class 11 · Mathematics · NCERT Mathematics
Straight Lines: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- Slope of a line
- Slope and angle
- Parallel lines
- Perpendicular lines
- Different forms of a line's equation
- General form
- Distance from a point
- Formulas and facts
- Slope formula
- Slope-intercept form
- Point-slope form
- Two-point form
- Intercept form
- General form
- Angle between two lines
- Worked examples
- Finding the equation of a line…
- Checking if two roads are…
- Distance from a village to a railway…
- Watch out for
- Forgetting that vertical lines have…
- Using the wrong sign in the…
- Confusing x-intercept and…
- Forgetting absolute value when…
- Mixing up which coordinate is…
What this chapter is about
This chapter introduces the study of straight lines in a two-dimensional coordinate plane. You learn how to describe a line using algebraic equations and how different forms of these equations reveal different properties of the line. Building on coordinate geometry from earlier classes, you now explore the relationship between geometry and algebra more deeply.
Key ideas
- Slope of a line measures its steepness. If a line passes through points (x₁, y₁) and (x₂, y₂), its slope m = (y₂ − y₁)/(x₂ − x₁), provided x₁ ≠ x₂. A horizontal line has slope 0; a vertical line has undefined slope.
- Slope and angle: If a non-vertical line makes an angle θ with the positive x-axis (measured anticlockwise), then m = tan θ. This angle θ is called the inclination of the line, where 0° ≤ θ < 180°.
- Parallel lines have equal slopes. If two lines have slopes m₁ and m₂, they are parallel when m₁ = m₂.
- Perpendicular lines satisfy the condition m₁ × m₂ = −1, meaning the product of their slopes equals negative one.
- Different forms of a line's equation suit different given information: slope-intercept form when slope and y-intercept are known, point-slope form when one point and slope are known, and so on.
- General form Ax + By + C = 0 (where A and B are not both zero) can represent any straight line, including vertical lines that other forms cannot easily express.
- Distance from a point to a line has a specific formula that uses the general form of the line's equation.
Formulas and facts to remember
- Slope formula: m = (y₂ − y₁)/(x₂ − x₁)Meaning: Rise divided by run between two points.
- Slope-intercept form: y = mx + cMeaning: m is the slope, c is the y-intercept (where the line crosses the y-axis).
- Point-slope form: y − y₁ = m(x − x₁)Meaning: The line with slope m passing through the point (x₁, y₁).
- Two-point form: (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁)Meaning: The line passing through two given points (x₁, y₁) and (x₂, y₂).
- Intercept form: x/a + y/b = 1Meaning: The line cutting the x-axis at (a, 0) and the y-axis at (0, b).
- General form: Ax + By + C = 0Meaning: The most general way to write any straight line.
- Angle between two lines: tan θ = |(m₁ − m₂)/(1 + m₁m₂)|, provided m₁m₂ ≠ −1.Meaning: Gives the acute angle between two non-perpendicular lines.
- Distance from point (x₁, y₁) to line Ax + By + C = 0: d = |Ax₁ + By₁ + C|/√(A² + B²)Meaning: The perpendicular (shortest) distance from the point to the line.
Common mistakes
- Forgetting that vertical lines have undefined slope → Use the form x = k for vertical lines; slope formulas do not apply to them.
- Using the wrong sign in the perpendicular condition, writing m₁ × m₂ = 1 → The correct condition is m₁ × m₂ = −1.
- Confusing x-intercept and y-intercept in the intercept form → In x/a + y/b = 1, the line crosses the x-axis at a and the y-axis at b.
- Forgetting absolute value when computing distance → The formula uses |Ax₁ + By₁ + C| because distance is always non-negative.
- Mixing up which coordinate is subtracted first in the slope formula → Be consistent: m = (y₂ − y₁)/(x₂ − x₁), same order in numerator and denominator.
Quick revision
- Slope m = (y₂ − y₁)/(x₂ − x₁) = tan θ, where θ is the inclination angle.
- Parallel lines: equal slopes. Perpendicular lines: product of slopes = −1.
- Point-slope form y − y₁ = m(x − x₁) is fastest when you know one point and the slope.
- Distance from (x₁, y₁) to Ax + By + C = 0 is |Ax₁ + By₁ + C|/√(A² + B²).
- Convert any line equation to general form Ax + By + C = 0 for uniform comparison.