# Coordinate Geometry

## 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.

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### 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 + CWhere,

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 = pwhere,

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 (x_{1}, y

_{1}) and (x

_{2}, y

_{2})

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 (x_{1},
y_{1}) and (x_{2}, y_{2}) 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 (x_{1}, y_{1}) is,

y - y_{1 }= m (x - x_{1})

where, m = slope

### Equation of line passing through two points

Equation of line passing through two points (x_{1}, y_{1})
and (x_{2}, y_{2}) is,

y - y_{1 }= \(\frac {y_2 - y_1}{x_2 - x_1}\) (x - x_{1})

### 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.

I \((\frac {ax_1+bx_2+cx_3}{a+b+c}, \frac {ay_1+by_2+cy_3}{a+b+c})\)### Circum-centre of triangle

The point of intersection of the perpendicular bisectors of the sides is called the circum-centre of the triangle.

### Ortho centre in triangle

The point of intersection of the altitudes is called the Ortho centre of the triangle.

**Note-**

Ortho centre in Right angle triangle-

### Ex-centre in Triangle

The circle which touches the triangle from the outside with all three sides, its centre is called the ex-centre.

I

_{1}\((\frac {-ax_1+bx_2+cx_3}{-a+b+c}, \frac {-ay_1+by_2+cy_3}{-a+b+c})\)

I_{2} \((\frac {ax_1-bx_2+cx_3}{a-b+c}, \frac {ay_1-by_2+cy_3}{a-b+c})\)

I_{3} \((\frac {ax_1+bx_2-cx_3}{a+b-c}, \frac {ay_1+by_2-cy_3}{a+b-c})\)

### Perpendicular distance between two parallel lines

If two parallel lines are ax + by + c_{1} = 0 and ax + by + c_{2}
= 0 then perpendicular distance between these two lines,

d = \(\left | \frac {c_1 - c_2}{\sqrt {a^2+b^2}} \right | \)

### Perpendicular distance of a line from a point

If point is (x_{1}, y_{1}) and line is ax + by + c = 0 then
perpendicular distance from the point,

d = \(\left | \frac {ax_1 + by_1 + c}{\sqrt {a^2+b^2}} \right | \)

### Angle between two lines

If slope of two lines m_{1} and m_{2} respectively
then the angle,

tan θ = \(\left | \frac {m_1 - m_1}{\sqrt {1+m_1m_2}} \right | \)

### Concurrent lines

If three equations a_{1}x + b_{1}y + c_{1} = 0,
a_{2}x + b_{2}y + c_{2} = 0 and a_{3}x +
b_{3}y + c_{3} = 0 are concurrent then

\[ \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix} =0 \]

### General quadratic equation

ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0

### General equation For circle

If h = 0 and a = b

x^{2} + y^{2} + 2gx + 2fy + c = 0

Centre (-g, -f)

Radius r = \(\sqrt {g^2 + f^2 - c}\)

#### Equation of circle central form

OP = r

\(\sqrt {(x-α)^2+(y-β)^2} = r\)

(x-α)^{2}+(y-β)^{2} = r^{2}

### Position of two circles and tangent

_{1},c

_{2}are the centre of the circle and r

_{1}, r

_{2}radius of the circle respectively then

#### case I-

If c_{1}c_{2} > r_{1} + r_{2}

_{}

Number of common tangents = 4

_{Case II-}

If c_{1}c_{2} = r_{1} + r_{2}

_{}

Number of common tangents = 3

#### case III-

If r_{1} - r_{2} < c_{1}c_{2} < r_{1} + r_{2}

Number of common tangents = 2

#### case IV-

If r_{1} - r_{2} = c_{1}c_{2}

Number of common tangents = 1

#### case V-

If c_{1}c_{2} < r_{1} - r_{2}

Number of common tangents = 0

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