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Coordinate Geometry Formula with Rules

Coordinate geometry

Also read,
Circle Formulas
Area Formulas
Lines and Angles
Straight Line
Conic Section Formula

Coordinate geometry:

To study coordinate geometry formula we must know about quadrants. As there are for quadrants each has its own sign rule.

All the four quadrants have their respective values with respective signs:

  • 1st Quadrant : (+x, +y)
  • 2nd Quadrant : (-x, +y)
  • 3rd Quadrant : (-x, -y)
  • 4th Quadrant : (+x, -y)

Distance between two points:

If A ( x1, y1) and B( x2, y2,), then distance d, from A to B =
\(\large d = \sqrt{(x_{2} -x_{1})^{2} + (y_{2} – y_{1})^{2}}\)

Midpoint of a line joining two points:

If A (x1, y1) and B( x2, y2,) represent the end point of line and M lies at midpoint of the line. Thus,

M (x, y) = \(\left ( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right )\)

Representation of Equation of line:

We can represent equation of a line in many ways, few of them is here-

  • General form- The general form of a line is given as Ax + By + C = 0.
  • Slope intercept form- Let us assume that x, y be the coordinate of a point which a line passes through, m be the slope of a line, and c be the y-intercept, then the equation of a line is given as- y=mx + c
  • Intercept form of a line- Let x-intercept and y-intercept of a line be a and b respectively, then the equation of a line is represented as-
    \(\large \boldsymbol{\frac{x}{a} + \frac{y}{b} = 1}\)

Coordinate geometry formula for slope of a line:

It can be obtained as per provided information as,
From General Form:
Ax+By+C=0 ⇒ By=−Ax–C

\(\large \Rightarrow y = -\frac{A}{B}x – \frac{C}{B}\)Let’s compare the above equation with y = mx + c,
we found that
\(\large \boldsymbol{m = -\frac{A}{B}}\)

Angle between two lines:

Let two lines be A and B, having their slopes to be m1 & m2 respectively.

Let θ be the angle between these two lines, then the angle between them can be represented as-
tan θ = \(\frac{m_{1} – m_{2}}{1 + m_{1} m_{2}}\)


When the two lines are parallel to each other then m1 = m2 = m

When the two lines are perpendicular to each other than
m1 . m2 = -1

Section Formula:

Section formula is a base part of coordinate geometry formula. In a line with two end points A and B having coordinates (x1,y1) and (x2, y2) Also M be any point collinear with the same line. The coordinates of M

Case 1.
When the ratio m:n is internally:

\(\large \left (\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n} \right )\)

When the ratio m:n is externally:

\(\large \left (\frac{mx_{2} – nx_{1}}{m – n}, \frac{my_{2} – ny_{1}}{m – n} \right )\)

Area of Triangle:

In coordinate geometry formula we also study about the area of the triangle. Thus the area of triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is:
\(\frac{1}{2}\) | X1 (Y­2 – Y3) + X2 (Y­3 – Y1) + X3 (Y­1 – Y2) |

If the area of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is zero, then the three points are collinear.

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