How far is the point from the line?
"How far is the point from the line?" is a free, interactive Class 11 Maths simulation for JEE, built on the lesson "Point-to-line distance" in Straight Lines. The blue line is 3x − 4y − 26 = 0. Drag the point P or use the sliders. The dashed orange segment PM meets the line at a right angle, and its length is the distance d = |3x − 4y − 26|/5. The distance of (x₁, y₁) from Ax + By + C = 0 is |Ax₁ + By₁ + C|/√(A² + B²). The expression is zero on the line and changes sign from one side to the other. JEE Main asked 42 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson Point-to-line distance in Straight Lines. Change the values and watch what happens.
In JEE Main
42 shifts analysed
42 questions from Straight Lines across the 42 JEE Main shifts Lumi has analysed (2017–2026).
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).
Questions
3 answers
- What does the "How far is the point from the line?" simulation show?
- The blue line is 3x − 4y − 26 = 0. Drag the point P or use the sliders. The dashed orange segment PM meets the line at a right angle, and its length is the distance d = |3x − 4y − 26|/5.
- Does JEE Main ask questions from Straight Lines?
- Yes. JEE Main asked 42 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Point-to-line distance" in Straight Lines (Class 11 Maths).
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