COORDINATE GEOMETRY
Definition:
A coordinate graph consists of a rectangular grid with two crossing lines called axes. The x-axis is the horizontal line and the y-axis is the vertical line. The axes intersect each other at the point (0,0) which is also called the origin.
Important Formulas
· Distance PQ = 
· Slope of PQ = m = 
· Equation of 
or y = mx + c
· The product of the slopes of two perpendicular lines is –1.
· The distance between the points (x1, y1) and (x2, y2) is 
· If point P(x, y) divides the segment AB, where A ≡ (x1, y1) and B ≡ (x2, y2), internally in the ratio m: n, then,
x= (mx2 + nx1)/(m+n)
and
y= (my2 + ny1)/(m+n)
· If P is the midpoint then, 
· If G (x, y) is the centroid of triangle ABC, A ≡ (x1, y1), B ≡ (x2, y2), C ≡ (x3, y3), then,
x = (x1 + x2 + x3)/3 and y = (y1 + y2 + y3)/3
· If I (x, y) is the in-centre of triangle ABC, A ≡ (x1, y1), B ≡ (x2, y2), C ≡ (x3, y3), then, 
where a, b and c are the lengths of the BC, AC and AB respectively.
· The equation of a straight line is y = MX + c, where m is the slope, and c is the y-intercept (tan θ = m, where θ is the angle that the line makes with the positive X-axis).
1. Distance Formula between A (x1, y1) & B (x2, y2).
Example: Find the distance between two points: A (2,3) and B(5,6)?
Using the above formula: 
The answer comes out to be 3& redic;2.
Position of a point in a plane
In coordinate geometry, points are placed on the "coordinate plane" as shown below. It has two scales - one running across the plane called the "x-axis" and another right angle to it called the y-axis. (These can be thought of as similar to the column and row in the paragraph above.) The point where the axes cross is called the origin and is where both x and y are zero.
On the x-axis, values to the right are positive and those to the left are negative. On the y-axis, values above the origin are positive and those below are negative. A point's location on the plane is given by two numbers; the first tells where it is on the x-axis and the second which tells where it is on the y-axis. Together, they define a single, unique position on the plane. So in the diagram above, the point A has an x value of 20 and a y value of 15. These are the coordinates of the point A, sometimes referred to as its "rectangular coordinates".
Distance between two points:
If A(x1,y1) and B (x2,y2) be two points, then
AB =√(x2-x1)2 + (y2-y1)2
Distance of a point from the origin:
The distance of a points A(x, y) from the origin O(0, 0) is given by
OA =√(x2+y2)
Area of a triangle:
If A(x1,y1), B(x2,y2) and C= (X3, Y3)be three vertices of a ∆ABC, then its area is given by:
∆ = 1/2 {x1(y2- Y3)+ x2(Y3- Y1) +X3(y1-y2)}
Condition of co linearity of three points:
Three points A(x1,y1), B(x2,y2) and C= (X3, Y3) are collinear if and only if ar(√ABC)= 0.
∴ A, B, C are collinear ⇒ x1(y2- Y3)+ x2(Y3- Y1) +x3(y1-y2) = 0
Division of a line segment by a point:
If a point p(x,y) divides the join of A(x1,y1) and B(x2,y2) in the ratio m:n, then
X= (mx2+nx1)/m+n and Y =(my2+ny1)/m+n
If A(x1,y1) and B(x2,y2) be the end points of a line segment AB, then the co-ordinates of midpoint of AB are
[(x1 + x2)/ 2 , (y1 + y2)/ 2]
Centroid of a triangle
The point of intersection of all the medians of a triangle is called its centroid. If A(x1,y1), B(x2,y2) and C= (X3, Y3) be the vertices of ABC, then the coordinates of its centroid are { (1/3 (x1+x2+x3),1/3 (y1+y2+Y3)}
Various types of Quadrilaterals
A quadrilateral is
A rectangle if its opposite sides is equal and diagonals are equal.
A parallelogram but not a rectangle, if it's opposite sides are equal and the diagonals are not equal.
A square, if all sides are equal and diagonals are equal.
A rhombus but not a square, if all sides are equal and diagonals are not equal.
Equations of lines
The equation of the x-axis is y =0.
The equation of y-axis is x = 0.
The equation of a line parallel to the y-axis at a distance from it is x= a.
The equation of a line parallel to the x-axis at a distance b from it is y= b.
The equation of a line passing through the points A(x1,y1) and B(x2,y2) is y-y1/ x-x1 = y2-y1/x2-x1. Slop of such a line is y2-y1/x2-x1.
The equation of a line in slope-intercept form is Y= mx+ c, where m is its slope.
Equation of a straight line
Slope Intercept form
Angle between two lines
Point slope form
Two-point form
Intercept form
Point of intersection of two lines
Circle
Q.4. Find the coordinate of the point which will divide the line joining the point (2,4) and (7,9) internally in the ratio 1:2?
A. (5/3, 1/3)
B. (3/8, 3/11)
C. (8/3, 11/3)
D. (11/3, 17/3)
Sol: Option D
The internal division will use the formula (mx2 + nx1)/(m + n)
y = (my2 + ny1)/(m + n).
So, the point becomes (11/3, 17/3).
Q.7. Find the equation of the line passing through (2, -1) and parallel to the line 2x – y = 4.
A. y = 2x – 5
B. Y=2x+6
C. Y=√2x +7
D. 4.y=2x+5
Sol: Option A
The given line is 2x – y = 4 ⇒ y = 2x – 4 (Converting into the form of y = mx + c)
Its slope = 2. The slope of the parallel line should also be 2.
Hence for the required line
m = 2 and (x1 , y1) = (2, -1).
Equation = y - y1 /x - x1 = y2 - y1 /x2 - x1
⇒ y - y1 /x - x1 = m
⇒ y – y1 = m (x – x1) ⇒ y – ( - 1) = 2 (x – 2)
⇒ y = 2x – 5.
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