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In two-dimensional coordinate geometry, each point in two-dimensional space is assigned with unique coordinates used to identify the point in the plane or graph.
Two Dimensions – Distance of a Point from a Line represents the distance (denoted by d) between any point P (x1, y1) and a line L whose equation is ax + by + c = 0, lying in a 2D plane.
Two Dimensions – Distance of a Point from a Line is given by the expression –
d = [|ax1 + by1 + c| / √ (a2 + b2)]. It represents the distance (denoted by d) between any point P (x1, y1) and a line L whose equation is ax + by + c = 0, lying in a 2D plane.
Introduction to the concept of Analytical Geometry
- Coordinate Geometry, also known as Analytical Geometry, is the field of mathematics that uses algebraic symbols and methods to represent and solve an equation for a problem in geometry.
- Applications of Analytical geometry:
- Its concept is used in physics, engineering, aviation, rocketry, space science, and spaceflight.
- It provides the foundation for most modern geometrical fields such as – algebraic geometry, differential geometry, discrete geometry, and computational geometry.
- Analytical geometry establishes a correspondence between geometric curves and algebraic equations, making it possible to reformulate problems in geometry in terms of equivalent problems in algebra and vice versa.
Introduction to the concept of Two-Dimensional Coordinate System
- In two-dimensional coordinate geometry, each point in two-dimensional space is assigned with unique coordinates used to identify the point in the plane or graph.
- Some important terms to be remembered are:
- Axes of Coordinates: The axes of coordinates are two intersecting straight lines used as reference lines in a graph or plane. For a plane with X and Y axes, and origin at O, OX and OY together form the axes of coordinates representing X-axis and Y-axis, respectively.
- Origin and its Coordinate: The point of intersection of the X and Y axes is known as the origin of that plane and is denoted by ‘O’. The coordinate of an origin in a cartesian plane is (0, 0).
- Abscissa and Ordinate: The distance of any point on a plane from the Y-axis and X-axis is respectively known as abscissa and ordinate.
- Quadrant: The X and Y axes divide the cartesian plane into 4 equal parts known as quadrants. A quadrant is one-fourth of the plane divided by coordinate axes.
Introduction to Two Dimensions – Distance of a Point from a Line.
- Two Dimensions – Distance of a Point from a Line represents the distance (denoted by d) between any point P (x1, y1) and a line L whose equation is ax + by + c = 0, lying in a 2D plane.
- The expression for two dimensions – distance of a point from a line in analytical geometry, with coordinates of any point P as (x1, y1) and equation of a line L as ax + by + c = 0, is given by –
d = [|ax1 + by1 + c| / √ (a2 + b2)].
Derivation of Two Dimensions – Distance of a Point from a Line.
Assume a line L represented by the equation Ax + By + C = 0, and a point N (x1, y1) lying in an XY Plane. The perpendicular distance of point N to line L is ‘d = NM’, which equals the perpendicular length drawn to line L. The x and y-intercepts for the given line are (-C/A) and (-C/B), respectively.
From the above figure, we can see that the line L meets the x and y axes at points B and A respectively and their coordinates are A (x3, y3) = A (0, -C/B) and B (x2, y2) B (-C/A, 0). The area of /\NAB is given by:
Area (/\NAB) = ½ x Base x Height = ½ (AB) (NM)
- NM = 2 [Area (/\ NAB)] – eq. 1
AB
We also know that area of /\NAB can be given by –
Area (/\NAB) = ½ | x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2) |
= ½ | x1 (0 + C/B) + (−C/A) (−C/B − y1) +0 (y1 − 0) |
= ½ | x1 C/B + y1 C/A + C2/AB |
= ½ |C/(AB) |. |Ax1 + By1 + C| – eq. 2
We know that distance AB = √ [(0 + C/A)2 + (C/B − 0)2] = |C| x √ [(1/A2) + (1/B2)]
= |C/AB|. √ (A2 + B2) -eq. 3
Putting values of eq. 2 and eq. 3 in eq. 1, we get,
NM = d = [|Ax1 + By1 + C| / √ (A2 + B2)], which is the two dimensions – distance of a point from a line in analytical geometry.
Conclusion:
Through this article on Analytical Geometry – Two Dimensions – Distance of a Point from a Line – Maths, students can understand and grasp the concepts of analytical or coordinate geometry and its applications, two-dimensional (or 2-D) coordinate system, as well as the formula and derivation of two dimensions – distance of a point from a line.
Two Dimensions – Distance of a Point from a Line represents the distance (denoted by d) between any point P (x1, y1) and a line L whose equation is ax + by + c = 0, lying in a 2D plane, and it is expressed as: d = [|ax1 + by1 + c| / √ (a2 + b2)].
