Coordinate Geometry - Finding Coordinates and Position of points

In this article, we will learn how to use the concepts of Coordinate Geometry in case of points.

Position of a point with respect to a line

We can easily find out whether given two points are on the same side of a line or on opposite sides. Let’s see how.

If A ( $x_{1}$ , $y_{1}$ ) and B ( $x_{2}$ , $y_{2}$ ) are two points and equation of a line is ax + by + c = 0, then:

Those two points will be on the same side of the line, if: Coordinate Geometry $(a x_{1} + b y_{1} + c)$ and $(a x_{2} + b y_{2} + c)$ have the same signs.

Those two points will be on the opposite sides of the line, if: Coordinate Geometry $(a x_{1} + b y_{1} + c)$ and $(a x_{2} + b y_{2} + c)$ have opposite signs.

Finding Coordinates of a point dividing a line

If we know the ratio in which a point divides a line internally or externally, we can find the coordinate of that point.

Internal division of a line segment

If a point A (x, y) divides the line joining two points P ( $x_{1}$ , $y_{1}$ ) and Q ( $x_{2}$ , $y_{2}$ ) internally, in the ratio m:n, then: Coordinate Geometry

x = $\frac{m x_{2} + n x_{1}}{m + n}$

y = $\frac{m y_{2} + n y_{1}}{m + n}$

Speacial Case

In case of the mid-point of a line segment, m:n = 1:1

So, coordinates of the mid-point of a line segment joining two points P ( $x_{1}$ , $y_{1}$ ) and Q ( $x_{2}$ , $y_{2}$ ), will be:

x = $\frac{x_{2} + x_{1}}{2}$

y = $\frac{y_{2} + y_{1}}{2}$

External division of a line segment

If a point A (x, y) divides the line joining two points P ( $x_{1}$ , $y_{1}$ ) and Q ( $x_{2}$ , $y_{2}$ ) externally, in the ratio m:n, then: Coordinate Geometry

x = $\frac{m x_{2} - n x_{1}}{m - n}$

y = $\frac{m y_{2} - n y_{1}}{m - n}$

Distance between two Coordinates

We can find distance between two points, if we know their coordinates.

If we have two points on a Cartesian plane, P ( $x_{1}$ , $y_{1}$ ) and Q ( $x_{2}$ , $y_{2}$ ), then: Coordinate Geometry

Distance between P and Q, d = $\sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}$

Special Case

Distance between a point P (x, y) and the origin (0, 0) = $\sqrt{(x - 0)^{2} + (y - 0)^{2}}$ = $\sqrt{x^{2} + y^{2}}$

Minimum distance of a point from a straight line

Minimum distance of a point from a straight line = Length of the perpendicular dropped from that point on that line. Coordinate Geometry

The length of perpendicular from a given point ( $x_{1}, y_{1}$ ) to a line ax + by + c = 0 can be calculated using the following formula:

d = $\frac{| a x_{1} + b y_{1} + c |}{\sqrt{a^{2} + b^{2}}}$

The length of perpendicular from the origin (0, 0) to a line ax + by + c = 0 is given by the following formula:

d = $\frac{| c |}{\sqrt{a^{2} + b^{2}}}$

Finding Collinear points

Three given points (say A, B and C) are collinear (i.e. they lie on the same straight line) if anyone of the following conditions is met:

Area of the ∆ABC is zero.
Slope of lines joining any two given points is the same. That is, Slope of AB = Slope of BC = Slope of CA.
If the sum of the lengths of any two line segments is equal to the third line segment. Say, if AB + BC = AC

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Last updated on Sep 23, 2021