Distance from a point to a line: Class 11 practice
The perpendicular distance from (x₁, y₁) to Ax + By + C = 0 is |Ax₁ + By₁ + C| ÷ √(A² + B²). Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Pythagorean triples
- Absolute value
The idea behind distance from a point to a line
The perpendicular distance from (x₁, y₁) to Ax + By + C = 0 is |Ax₁ + By₁ + C| ÷ √(A² + B²). Between parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 it is |C₁ − C₂| ÷ √(A² + B²).
Step-by-step method
- Write the line as Ax + By + C = 0.
- Substitute the point into the numerator.
- Divide by √(A² + B²), keep the absolute value.
Worked example 1 (Basic)
Find the distance of (1, 3) from 6x + 8y − 12 = 0.
- |6(1) + 8(3) − 12| = 18
- ÷ √(100) = 10
- = 9/5
- Answer: 9/5
Worked example 2 (Standard)
Find the distance of (2, -5) from 6x + 8y + 19 = 0.
- |6(2) + 8(-5) + 19| = 9
- ÷ √(100) = 10
- = 9/10
- Answer: 9/10
Worked example 3 (Challenge)
Find the distance between 3x + 4y + 8 = 0 and 3x + 4y + 18 = 0.
- |8 − 18|/√(9 + 16) = 10/5
- = 2
- Answer: 2
Common mistakes and misconceptions
- Dropping the absolute value
- Not rearranging to = 0 first
Practice questions
- (Basic) Find the distance of (3, -4) from 12x + 5y + 18 = 0.
- (Basic) Find the distance of (-3, -2) from 5x + 12y − 3 = 0.
- (Standard) Find the distance of (-5, -4) from 4x + 3y + 16 = 0.
- (Standard) Find the distance of (1, -3) from 3x + 4y − 7 = 0.
- (Challenge) Find the distance between 5x + 12y − 7 = 0 and 5x + 12y + 32 = 0.
- (Challenge) Find the distance between 4x + 3y − 5 = 0 and 4x + 3y + 0 = 0.
Answer key with full working
- Q1: |12(3) + 5(-4) + 18| = 34 → ÷ √(169) = 13 → = 34/13 → Answer: 34/13
- Q2: |5(-3) + 12(-2) − 3| = 42 → ÷ √(169) = 13 → = 42/13 → Answer: 42/13
- Q3: |4(-5) + 3(-4) + 16| = 16 → ÷ √(25) = 5 → = 16/5 → Answer: 16/5
- Q4: |3(1) + 4(-3) − 7| = 16 → ÷ √(25) = 5 → = 16/5 → Answer: 16/5
- Q5: |-7 − 32|/√(25 + 144) = 39/13 → = 3 → Answer: 3
- Q6: |-5 − 0|/√(16 + 9) = 5/5 → = 1 → Answer: 1
Syllabus note
This skill is listed under "Straight Lines" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is distance from a point to a line in?
"Straight Lines", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.