Coordinate Geometry
 The branch of geometry that sets up a definite correspondence between the position of a point in a plane and a pair of algebraic numbers called coordinates is called Coordinate geometry.
 The distance of a point from the Yaxis is called the Xcoordinate or Abscissa.
 The distance of a point from the Xaxis is called the Ycoordinate or ordinate.
 The coordinates of a point on the Xaxis are of the form (x, 0) and of a point on the Yaxis are of the form (0, y).
 The abscissa and ordinate of a point taken together is known as coordinates of a point.
 The point of intersection of the axes of coordinates is called the origin.
 The quarter plane that results from the division of the plane by the coordinate axes is called a quadrant.
 Signs of x and y coordinates in the four quadrants:
Quadrant

Xcoordinate

ycoordinate

Sign of the coordinates of the point

I

Positive

Positive

\[(+,+)\]

II

Negative

Positive

\[(,+)\]

III

Negative

Negative

\[(,)\]

IV

Positive

Negative

\[(+,)\]

Distance Formula:
 The distance between two points \[A({{x}_{1}},{{y}_{1}})\] and \[B({{y}_{2}},{{y}_{2}})\] is given by
\[AB=\sqrt{{{({{x}_{2}}{{x}_{1}})}^{2}}+{{({{y}_{2}}{{y}_{1}})}^{2}}}\]
 The distance of the point \[p(x,y)\]from the origin 0 (0,0) is given by \[OP=\sqrt{{{x}^{2}}+{{y}^{2}}}\].
 Properties of various types of quadrilaterals: a quadrilateral is a
(a) Rectangle: If its opposite sides are equal and the diagonals are equal.
(b) Square: If all its sides are equal and the diagonals are equal.
(c) Parallelogram: If its opposite sides are equal.
(d) Parallelogram but not a rectangle: If its opposite sides are equal and the diagonals are not equal.
(e) Rhombus but not a square: If all its sides are equal and the diagonals are not equal.
 Collinear points: Points are said to be collinear if they lie on the same straight line.
 Test for co linearity of given points: Given three points A, B and C, find the distances AB, BC and AC. If the sum of any two of these distances is equal to the third distance, then the given points are collinear.
 Section formula: The coordinates of the point \[p(x,y)\] which divides the line segment joining \[A({{x}_{1}},{{y}_{1}})\]and \[B({{x}_{2}},{{y}_{2}})\]internally in the ratio m : n are given by \[x=\frac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\]and \[y=\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\]
Note: The coordinates of a point \[p(x,y)\]which divides the line segment joining \[A({{x}_{1}},{{y}_{1}})\]and\[B({{x}_{2}},{{y}_{2}})\]externally in the ratio m: n are given by \[x=\frac{m{{x}_{2}}n{{x}_{1}}}{m+n}\]and \[y=\frac{m{{y}_{2}}n{{y}_{1}}}{mn}\].
 Midpoint formula: The coordinates of the midpoint M of a line segment AB joining the points \[A({{x}_{1}},{{y}_{1}})\]and\[B({{x}_{2}},{{y}_{2}})\]are \[M\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2} \right)\]
 Centroid: The point of intersection of the medians of a triangle is called its centroid. The centroid of a triangle divides each median in the ratio 2:1.
 The coordinates of the centroid of a triangle are \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\]
 Area of a triangle: If \[A({{x}_{1}},{{y}_{1}}),B({{x}_{2}},{{y}_{2}})\]and \[C({{x}_{3}},{{y}_{3}})\]are the vertices of a triangle, its area is given by\[\frac{1}{2}{{x}_{1}}({{y}_{2}}{{y}_{3}})+{{x}_{2}}({{y}_{3}}{{y}_{1}})+{{x}_{3}}({{y}_{1}}{{y}_{2}})\]
 If the area of triangle is 0, then the three points that form its vertices are collinear. In other words, three points are collinear if \[{{x}_{1}}({{y}_{2}}{{y}_{3}})+{{x}_{2}}({{y}_{3}}{{y}_{1}})+{{x}_{3}}({{y}_{1}}{{y}_{2}})=0\].
 Incentre: The points of concurrence of the bisectors of the angles of a triangle are called the incentre of the triangle. The coordinates of the incentre of a triangle are \[\left( \frac{a{{x}_{1}}+b{{x}_{2}}+c{{x}_{2}}}{a+b+c},\frac{a{{y}_{1}}+b{{y}_{2}}+c{{y}_{3}}}{a+b+c} \right)\]where a, b and c are the sides of\[\Delta \text{ABC}\].