# Coordinate Geometry

## Coordinate Geometry functions and formulas or Identity list

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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 + 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 - y= 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 - y= $$\frac {y_2 - y_1}{x_2 - x_1}$$ (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.

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.

Note-

Circum centre in Right angle triangle-

O $$(\frac {x_1+x_3}{2}, \frac {y_1+y_3}{2})$$

### 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.

I1 $$(\frac {-ax_1+bx_2+cx_3}{-a+b+c}, \frac {-ay_1+by_2+cy_3}{-a+b+c})$$

I2 $$(\frac {ax_1-bx_2+cx_3}{a-b+c}, \frac {ay_1-by_2+cy_3}{a-b+c})$$

I3 $$(\frac {ax_1+bx_2-cx_3}{a+b-c}, \frac {ay_1+by_2-cy_3}{a+b-c})$$

### Perpendicular distance between two parallel lines

If two parallel lines are ax + by + c1 = 0 and ax + by + c2 = 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 (x1, y1) 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 m1 and m2 respectively then the angle,

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

### Concurrent lines

If three equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 and a3x + b3y + c3 = 0 are concurrent then

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

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0

### General equation For circle

If h = 0 and a = b

x2 + y2 + 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 = r2

### Position of two circles and tangent

Let c1,c2 are the centre of the circle and r1, r2 radius of the circle respectively then

#### case I-

If c1c2 > r1 + r2

Number of common tangents = 4

#### Case II-

If c1c2 = r1 + r2

Number of common tangents = 3

#### case III-

If r1 - r2 < c1c2 < r1 + r2

Number of common tangents = 2

#### case IV-

If r1 - r2 = c1c2

Number of common tangents = 1

#### case V-

If c1c2 < r1 - r2

Number of common tangents = 0