Understanding the Distance of Point ( P(0,0) ) from the Line ( 2x 3y 0 )

In this article, we will delve into the geometric concept of the distance of a point from a line. Specifically, we will explore the distance of the point ( P(0,0) ) from the line given by the equation ( 2x 3y 0 ). This concept is fundamental in understanding various applications in mathematics and its related fields.

1. Introduction to Line Equations and Points

A line in the Cartesian plane is usually represented by the general form of the equation ( Ax By C 0 ), where ( A, B, ) and ( C ) are constants. The point ( P(x, y) ) is a location in the plane, and the concept of distance from a point to a line is an important one in geometry and algebra.

2. The Specific Case: Point ( P(0,0) )

Consider the line described by the equation ( 2x 3y 0 ). Let's examine the distance from the origin, represented by the point ( P(0,0) ), to this line.

2.1 Geometric Interpretation

Imagine standing at the origin ( P(0,0) ) and looking towards the line ( 2x 3y 0 ). The origin is exactly on the line because substituting ( x 0 ) and ( y 0 ) into the equation yields ( 2(0) 3(0) 0 ), which is true. Therefore, the distance from the point ( P(0,0) ) to the line ( 2x 3y 0 ) is zero.

2.2 Practical Interpretation

Using a more practical analogy, suppose you are standing on a riverbank and someone asks, "What is your distance from the river?" If you are standing directly on the riverbank, your distance is zero. Similarly, the point ( P(0,0) ) is directly on the line ( 2x 3y 0 ). Thus, the distance is zero.

3. Formal Derivation: Distance from a Point to a Line

The perpendicular distance ( d ) from a point ( (x_1, y_1) ) to a line ( Ax By C 0 ) is given by the formula:

$$ d frac{|Ax_1 By_1 C|}{sqrt{A^2 B^2}} $$

3.1 Applying the Formula

Now, let's apply this formula to the specific case of the point ( P(0,0) ) and the line ( 2x 3y 0 ).

For the point ( P(0,0) ), ( x_1 0 ) and ( y_1 0 ). The line equation is ( 2x 3y 0 ), where ( A 2 ), ( B 3 ), and ( C 0 ).

Substituting these values into the formula:

$$ d frac{|2(0) 3(0) 0|}{sqrt{2^2 3^2}} $$

$$ d frac{0}{sqrt{4 9}} $$

$$ d frac{0}{sqrt{13}} $$

$$ d 0 $$

4. Conclusion

In conclusion, the distance from the point ( P(0,0) ) to the line ( 2x 3y 0 ) is zero. This result is consistent with both geometric and practical interpretations. Understanding the concept of the distance from a point to a line is crucial for various applications in mathematics, including geometry, algebra, and calculus.

5. Related Topics and Further Reading

For further exploration of this topic, consider studying:

The parametric form of a line The intersection of two lines The concept of perpendicularity in geometry

5.1 Practical Applications

This concept has practical applications in fields such as robotics, computer graphics, and engineering, where understanding distances between points and lines is crucial.

5.2 Key Takeaways

The distance from point ( (0,0) ) to the line ( 2x 3y 0 ) is zero. The origin is directly on the line ( 2x 3y 0 ).