Coordinate geometry:

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

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

• 1^st Quadrant : (+x, +y)
• 2^nd Quadrant : (-x, +y)
• 3^rd Quadrant : (-x, -y)
• 4^th 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 = \(\frac{m_{1} – m_{2}}{1 + m_{1} m_{2}}\)

Case1.

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

Case2.
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:
\(\large \left (\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n} \right )\)

Case2.
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 (Y_­2 – Y₃) + X₂ (Y_­3 – Y₁) + X₃ (Y_­1 – 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.