How to Find the Distance of a Point from a Line? Stepwise Method & Examples
The concept of distance of a point from a line is an essential idea in coordinate geometry, used to calculate the shortest path between a point and a line. Understanding this distance helps in solving problems in mathematics, physics, engineering, and various competitive exams. Let’s explore the meaning, formula, derivation, worked examples, and related tips for this topic, following Vedantu’s high academic standards.
Understanding Distance of a Point from a Line
A distance of a point from a line refers to the shortest, or perpendicular, distance from a given point to a given straight line. This concept is widely used in coordinate geometry, analytic geometry, and vector algebra. The shortest distance is always measured along the line passing through the point at a right angle (perpendicular) to the given line.
Formula Used in Distance of a Point from a Line
The standard formula used is:
\( d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \)
where:
- \( (x_1, y_1) \) are the coordinates of the point
- \( Ax + By + C = 0 \) is the equation of the line.
Here’s a helpful table to understand distance of a point from a line more clearly in different scenarios:
Distance of a Point from a Line Table
| Form | Formula | Dimension |
|---|---|---|
| Cartesian (2D) | \( d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \) | 2D |
| Vector form | \( d = \frac{|(\vec{a} - \vec{b}) \cdot \vec{n}|}{|\vec{n}|} \) | 2D / 3D |
| 3D (vector approach) | \( d = \frac{|(\vec{AP} \times \vec{AB})|}{|\vec{AB}|} \) | 3D |
This table shows the popular forms used to compute the distance of a point from a line, depending on context and dimensions.
Derivation of the Formula (Coordinate Geometry Approach)
Let’s derive the formula for the perpendicular distance of a point \( (x_1, y_1) \) from the line \( Ax + By + C = 0 \):
1. The area of triangle formed by the point and two points on the line can be found using coordinates.2. Express the height (distance) in terms of area and base of the triangle.
3. Use the area formula for triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \):
4. Substitute coordinates of intersection points of the line with axes and the point \( (x_1, y_1) \).
5. After algebraic simplification, arrive at:
Worked Example – Solving a Problem
Let’s apply the formula step-by-step on a sample problem:
Example: Find the distance of the point (5, 1) from the line y = 3x + 1.
1. Rewrite the line in standard form:2. Identify A, B, C, and the point:
3. Plug into the distance formula:
4. Simplify numerator:
5. Final answer:
Common Mistakes to Avoid
- Missing the modulus/absolute value in the numerator.
- Not converting the line to the standard form \( Ax + By + C = 0 \) before using the formula.
- Confusing between the perpendicular distance formula and distance between two points.
Real-World Applications
The distance of a point from a line is widely used in navigation, construction (to find shortest path to a boundary), robotics, physics, and computer graphics. It is also crucial for board exams, engineering entrances, and practical geometry problems. Vedantu explains this with simple visuals and everyday contexts, making the concept relatable and easy to apply.
Key Differences: Line vs. Line Segment and More
| Case | Distance Type | Formula Used? |
|---|---|---|
| Infinite line | Shortest (perpendicular) distance | Yes, main formula applies |
| Line segment | May be endpoint or perpendicular (depends on projection) | Check if foot of perpendicular lies on segment |
| Parallel lines | Constant shortest distance | Special form of the formula |
Always identify if the question is about a whole line, a finite segment, or parallel lines to apply the correct method.
Page Summary
We have explored the idea of distance of a point from a line, learned how to use and derive the formula, solved stepwise problems, seen common mistakes, and understood where it is used in daily life and exams. Practice more with Vedantu to be confident using these formulas.
Explore Related Topics
- Equation of a Line – Understand line representations vital to distance problems.
- Perpendicular Distance of a Point from a Plane – Extend this concept into 3D space for advanced questions.
- Distance Between Two Points – Build a foundation for all distance-based calculations.
- Line Segment – Know the difference between a segment and an infinite line in geometry.
- Angle Between a Line and a Plane – Explore more on 3D concepts related to lines and distances.
- Coordinate Geometry – Strengthen your skills with related coordinate-based chapters.
- Three Dimensional Geometry – Dive into 3D geometry for competitive exam prep.
- Vector Algebra for Class 12 – Apply the distance formula using vectors for advanced problems.
- Straight Lines – Master the theory of straight lines for clarity in distance calculations.
- Cartesian Coordinates – Brush up the basics of points, axes, and coordinates in geometry.
FAQs on Distance of a Point from a Line – Concepts & Formula
1. What is the formula for the distance of a point from a line?
The **distance** (d) of a point (x₁, y₁) from a line represented by the equation Ax + By + C = 0 in 2D coordinate geometry is given by the formula:
d = \frac{|Ax₁ + By₁ + C|}{\sqrt{A^2 + B^2}}.
This formula calculates the shortest, **perpendicular distance** between the point and the line.
2. How do you find the distance of a point from a line in 3D?
To find the **distance of a point from a line in 3D**, use vector algebra:
1. Represent the line with a point A and direction vector \vec{d}.
2. Find the vector \vec{AP} from point A on the line to the given point P.
3. Use the formula:
d = \frac{||\vec{AP} \times \vec{d}||}{||\vec{d}||}, where \times denotes the cross product.
This formula gives the shortest distance from the point to the line in space.
3. Is the shortest distance always perpendicular?
Yes, the **shortest distance** from a point to a line is always along the **perpendicular** drawn from the point to the line. This is because the perpendicular segment is the minimal possible distance compared to any other oblique path connecting the point and the line.
4. How is the distance formula used in vector form?
In **vector form**, to find the distance between a point P and a line passing through point A with direction vector \vec{d}, calculate the vector \vec{AP} = \vec{P} - \vec{A}. Then, the distance is the magnitude of the component of \vec{AP} perpendicular to \vec{d}:
d = \frac{||\vec{AP} \times \vec{d}||}{||\vec{d}||}.
This method uses **cross product** and vector norms to find the perpendicular distance in coordinate geometry.
5. Can I use a calculator to solve distance problems?
Yes, using a **distance of a point from a line calculator** can help quickly solve problems, especially those involving decimals or complex numbers. However, understanding the underlying **formula, derivation, and stepwise method** is essential for exams and conceptual clarity.
6. What is the distance between a point and a line segment?
The **distance between a point and a line segment** differs from that to an infinite line. If the perpendicular from the point falls outside the segment, the shortest distance is to the nearest endpoint. Otherwise, it’s the perpendicular (shortest) distance to the segment's line. This distinction is important in geometry and real-life applications.
7. Why do students confuse the line formula with segment calculation?
Students often confuse the formula because the **distance of a point from a line** assumes the line is infinite, whereas line **segments have endpoints**. Misapplying the infinite line formula to segments can lead to incorrect answers. Understanding when to check if the foot of the perpendicular lies within the segment is crucial.
8. What errors cause marks lost in distance-of-point-from-line questions?
Common errors include:
• Incorrect substitution of point or line coefficients in the formula.
• Forgetting to take the absolute value in the numerator.
• Miscalculating or ignoring the square root in the denominator.
• Confusing lines with line segments or points lying outside the segment.
Careful stepwise calculation and verifying results help avoid these pitfalls.
9. How is the 3D formula different from the 2D one?
The **2D distance** formula uses line equation coefficients **A, B, C** and point coordinates to find perpendicular distance. In **3D**, since a line has direction vectors, the distance formula involves the **cross product** of vectors to find the shortest distance from the point to the line in space. Thus, 3D distance is vector-based rather than algebraic-coefficient based.
10. Why is direction important in vector solutions?
Direction vectors define the orientation of the line in space. In vector solutions, the **direction determines how the perpendicular or shortest distance is calculated via vector projections or cross products**. Without direction, distance computation between a point and a line in 3D is impossible because the line's position is undefined.
11. How to quickly check if your distance calculation is correct?
To quickly verify accuracy:
• Substitute the point coordinates into the line equation to check if it satisfies the equation.
• For 2D, compute the perpendicular foot coordinates and confirm the right angle.
• Compare your answer to a rough estimate or graph.
This check helps catch substitution or arithmetic errors before final submission.
12. What real-world applications use this concept?
The concept of **distance of a point from a line** is applied in:
• Engineering designs to ensure components maintain specific clearances.
• Navigation and GPS for shortest path calculations.
• Physics problems involving forces and trajectories.
• Computer graphics for rendering and object collision detection.
These examples show its practical significance beyond academic exams.
