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 plane

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).

Coordinate plane

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

line equation
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

line
y = mx + C

Where,

m(slope) = tan θ 

C = Intersection cut on Y-axis

iii. Intercept form

line

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

where,

a = Intersection cut on X-axis

b = Intersection cut on Y-axis

iv. Normal Form

line
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-

formula

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

ii. External Division-

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

Mid-Point Formula 

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.

coordinate geometry

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 equal distance from the sides of the triangle.

coordinate geometry
I \((\frac {ax_1+bx_2+cx_3}{a+b+c}, \frac {ay_1+by_2+cy_3}{a+b+c})\)

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

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

General quadratic equation

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

circle

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

circle

Number of common tangents = 4

Case II-

If c1c2 = r1 + r2

circle

Number of common tangents = 3

case III- 

If r1 - r2 < c1c2 < r1 + r2

circle

Number of common tangents = 2

case IV-

If r1 - r2 = c1c2 

circle

Number of common tangents = 1

case V-

If c1c2 < r1 - r2 

circle

Number of common tangents = 0