Simulation · Maths · Class 11

How far is the point from the line?

From the lesson Point-to-line distance in Straight Lines. Change the values and watch what happens.

The idea behind it

NCERT §9.4

How far a point is from a line means the length of the perpendicular dropped from the point onto the line. No other path from the point to the line is shorter.
The distance of P(x₁, y₁) from the line Ax + By + C = 0 is d = |Ax₁ + By₁ + C|/√(A² + B²). It comes from writing twice the area of the triangle PQR, where Q and R are the points where the line meets the axes, as the base QR times the height PM.
The point (3, −5) is at distance |9 + 20 − 26|/√(9 + 16) = 3/5 from the line 3x − 4y − 26 = 0. The origin is at distance |−10|/5 = 2 from 3x + 4y − 10 = 0.
Putting the point into Ax + By + C gives zero exactly when the point lies on the line; the sign of the value tells which side of the line the point is on.
A distance measured along a given direction is not the perpendicular distance. From P(4, 1), along a line of slope −1, the line 4x − y = 0 is reached at (1, 4), at a distance 3√2.
The mirror image of a point in a line is found from two facts: the line joining the point and its image is perpendicular to the mirror line, and its mid-point lies on it. The image of (1, 2) in x − 3y + 4 = 0 is (6/5, 7/5).

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