Distance and Family of Lines
The distance from a point to ax + by + c = 0 is |ax0 + by0 + c| / √(a2 + b2), and any line through the intersection of two lines is a linear combination of them.
What is Distance and Family of Lines?
The distance from a point to ax + by + c = 0 is |ax0 + by0 + c| / √(a2 + b2), and any line through the intersection of two lines is a linear combination of them.
Key formula / rule: Distance between parallel lines uses the same formula with matched coefficients.
Key points
- Compute the perpendicular distance from a point to a line.
- Apply the family-of-lines method to concurrency problems.
Common exam trap
Forgetting the absolute value and reporting a negative distance.
Definitions
- Term
Distance and Family of Lines
- Meaning
The distance from a point to ax + by + c = 0 is |ax0 + by0 + c| / √(a2 + b2), and any line through the intersection of two lines is a linear combination of them.
- Term
Distance and Family of Lines — explanation
- Meaning
The distance formula converts geometric conditions into algebra, while the family-of-lines idea removes the need to compute intersection points explicitly.
Learning objectives
Compute the perpendicular distance from a point to a line.
Apply the family-of-lines method to concurrency problems.
Formulae
- Key point
Distance between parallel lines uses the same formula with matched coefficients.
- Key point
L1 + k L2 = 0 represents the family through the intersection of L1 and L2.
- Key point
The angle bisectors of two lines are found by equating the distance expressions.
Prerequisites
MAT-U3-SL-T1-S1-C1
Common mistakes
Forgetting the absolute value and reporting a negative distance.
Using the family formula with unmatched coefficient normalisation.
Keywords
Distance
Family
Lines
Practice preview
The equation of the line passing through the intersection of the lines 2x + 3y - 4 = 0 and x - 5y + 7 = 0 and also passing through the point (2, 1) is:…
easy
Find the value of 'k' such that the point (1, 2) is equidistant from the lines 3x + 4y - 1 = 0 and 8x + 6y - k = 0.…
medium
A point on the line x + y = 4 which is at a unit distance from the line 4x + 3y - 10 = 0 is:…
medium