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: $(ax_1 + by_1 + c)$ and $(ax_2 + by_2 + c)$ have the same signs.

Those two points will be on the opposite sides of the line, if: $(ax_1 + by_1 + c)$ and $(ax_2 + by_2 + c)$ 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:

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:

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:

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.

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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