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4.Distance of a Point from a Line
Class 11 • Mathematics • NCERT
Distance of a Point from a Line
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This lesson explains how to find the shortest distance between a point and a straight line using the standard NCERT formula. Students learn the concept clearly with simple, exam-oriented examples.
Lesson Objectives
- Understand the meaning of distance from a point to a line.
- Use the standard distance formula.
- Apply the formula to NCERT-style problems.
- Avoid common exam mistakes.
1. Meaning of Distance from a Point to a Line
The distance of a point from a line is the length of the perpendicular drawn from the point to the line.
The shortest distance between a point and a line is always along the perpendicular.
2. Standard Form of Line
To apply the distance formula, the equation of the line must be in standard form.
Standard form:
ax + by + c = 0
ax + by + c = 0
3. Distance Formula (NCERT)
The distance of the point (x₁, y₁) from the line ax + by + c = 0 is:
Distance = |ax₁ + by₁ + c| / √(a² + b²)
4. Important NCERT Notes
Some important points to remember while using the formula:
• Take absolute value in the numerator
• Line must be in ax + by + c = 0 form
• Distance is always positive
• Line must be in ax + by + c = 0 form
• Distance is always positive
5. Simple NCERT Examples
• Find the distance of point (1,2) from line 3x + 4y − 5 = 0
Distance = |3(1) + 4(2) − 5| / √(3² + 4²)
= |6| / 5 = 6/5
Distance = |3(1) + 4(2) − 5| / √(3² + 4²)
= |6| / 5 = 6/5
Practice Questions (NCERT)
- Define distance of a point from a line.
- Write the standard form of a straight line.
- State the distance formula.
- Why is absolute value used in the formula?
- Find the distance of (0,0) from line x + y − 4 = 0.
- Is distance ever negative?
- What is the denominator in the distance formula?
- Why must the line be in standard form?
- What type of distance is measured?
- Is this topic part of NCERT Class 11 syllabus?
✅ Show Answer Key
- The length of the perpendicular from the point to the line.
- ax + by + c = 0
- |ax₁ + by₁ + c| / √(a² + b²)
- To make distance positive.
- 4/√2 = 2√2
- No
- √(a² + b²)
- Formula is defined for that form.
- Shortest (perpendicular) distance.
- Yes
© Aviate Learning – Straight Lines (NCERT Class 11)