Coordinate Geometry

July 22, 2022 October 09, 2022

Coordinate Geometry functions and formulas or Identity list

This is for all competitive exams like SSC, Railway, Bank, TGT, PGT and other competitive exams and academic exams like intermediate or High school exam CBSE, ICSE (class 10, 11, 12) or State board.

UPI ID:- achalup41-1@oksbi

Contact Email ID:- achalup41@gmail.com

Coordinate Plane or Cartesian plane

It helps to locate any point in 2-D space. Some essential terms are given below.

Origin- It is the centre point of the cartesian plane and it is generally denoted by 'O' and coordinate (0, 0).

Abscissa- It is the value of distance on the X-axis. It denotes value from the origin negative and positive sides on the X-axis.

Ordinate- It is the value of distance on the Y-axis. It denotes value from the origin negative and positive sides on the Y-axis.

Cartesian Coordinate

It is in the form (x, y).

e.g.- (2, 3), (3, 4), (5, 8) e.t.c

Polar Coordinate

It is in the form (r, θ).

Where,

cos θ = \(\frac {x}{r}\) ⇒ x = r cos θ

sin θ = \(\frac {y}{r}\) ⇒ y = r sin θ

tan θ = \(\left|\frac{y}{x}\right|\) ⇒ θ = \(tan^{-1}\left|\frac{y}{x}\right|\)

r = \( \sqrt {x^2 + y^2}\)

e.g.- \((2, \frac {π}{2})\), \((5, \frac {π}{6})\)), (8, 60°), (3, 45°) e.t.c

Distance formula

Cartesian form

Let the two-point P(x, y) and Q(x, y) then the distance between PQ.

PQ = \(\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Polar form

PQ = \(\sqrt {(r_1)^2 + (r_2)^2 - 2r_1r_2cos (θ_1 - θ_2)}\)

Straight Line Equations

i. General form

ax + by + c = 0

👉 Slope of the equation = \(-\frac {coefficient \space of \space 'a'}{coefficient \space of \space 'b'}\) = \(-\frac {a}{b}\)

ii. Slope form

y = mx + C

Where,

m(slope) = tan θ

C = Intersection cut on Y-axis

iii. Intercept form

\(\frac {x}{a} + \frac {y}{b} = 1\)

where,

a = Intersection cut on X-axis

b = Intersection cut on Y-axis

iv. Normal Form

x cos α + y sin α = p

where,

p = The length of the perpendicular 'OM' from the origin to the line.

α = Angle between normal (OM) and X-axis.

Slope of a line

The formula of the slope if a line passes through two points (x₁, y₁) and (x₂, y₂)

m = \(\frac {y_2 - y_1}{x_2 - x_1}\)

Section Formula or Division formula

A point (x, y) divides the line segment by joining the points (x₁, y₁) and (x₂, y₂) in the ratio m : n then,

i. Internal Division-

p (x, y) = \((\frac {mx_2 + nx_1}{m+n}, \frac {my_2 + ny_1}{m+n})\)

ii. External Division-

p (x, y) = \((\frac {mx_2 - nx_1}{m-n}, \frac {my_2 - ny_1}{m-n})\)

Mid-Point Formula

p (x, y) = \((\frac {x_2 + x_1}{2}, \frac {y_2 + y_1}{2})\)

Equation of line passing through a point

Equation of line passing through the point (x₁, y₁) is,

y - y₁= m (x - x₁)

where, m = slope

Equation of line passing through two points

Equation of line passing through two points (x₁, y₁) and (x₂, y₂) is,

y - y₁= \(\frac {y_2 - y_1}{x_2 - x_1}\) (x - x₁)

Centroid of Triangle

The point of intersection of the medians of a triangle is called the centroid. The centroid divides the median in the ratio 2 : 1 internally.

G \((\frac {x_1+x_2+x_3}{3}, \frac {y_1+y_2+y_3}{3})\)

In-centre of triangle

The point of intersection of the interior angle bisectors of a triangle is called the In-centre and it is located at an equal distance from the sides of the triangle.