1 STATISTICS Data Sufficiency Set - 1 [Quiz]
20 Min
2 STATISTICS Data Sufficiency Set - 2 [Quiz]
20 Min
3 STATISTICS Data Sufficiency Set - 3 [Quiz]
20 Min
4 STATISTICS Data Sufficiency Set - 4 [Quiz]
2 Min
5 Coordinate Geometry
N/A
It is a system of geometry, where the position of points on the plane is described by using an ordered pair of numbers.
Rectangular Coordinate Axes
The lines XOX' and YOY' are mutually perpendicular to each other and they meet at point O which is called the origin.
Line XOX' represents X-axis and line YOY' represents Y-axis and together taken, they are called coordinate axes.
Any point in coordinate axis can be represented by specifying the position of x and y-coordinates
Quadrants
The X and Y-axes divide the cartesian plane into four regions referred to quadrants
- The coordinates of point O (origin) are taken as (0,0).
- The coordinates of any point on X-axis are of the form (x, 0).
- The coordinates of any point on Y-axis are of the form (0, y)
Formulae:
Distance Formula
Distance between Two Points If A (x1, y1) and B (x2, y2) are two points, then ^2+\left(y_2-y_1\right)^2} "AB=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}")
Distance of a Point from the Origin
The distance of a point A (x, y) from the origin O (0, 0) is given by 
Area of triangle
If A (x1, y1) B (x2, y2) and C (x3, y3) are three vertices of a Triangle ABC, then its area is given by
Area of triangle 
(x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2))
Collinearity of Three Points
Three points A (x1, y1) B (x2, y2) and C (x3, y3) are collinear, if
(i) Area of triangle ABC is 0
(ii) Slope of AB = Slope of BC = Slope of AC
(iii) Distance between A and B + Distance between B and C = Distance between A and C
Centroid of a Triangle
Centroid is the point of intersection of all the three medians of a triangle. If A (x1, y1) B (x2, y2) and C (x3, y3) are the vertices of triangle ABC, then the coordinates of its centroid are
Circumcentre
The circumcentre of a triangle is the point of inter section of the perpendicular bisectors of its sides and is equidistance from all three vertices.
If A (x1, y1) B (x2, y2) and C (x3, y3) are the vertices of triangles and O (x, y) is the circumcentre of triangle ABC, then OA = OB= OC
^2+\left(y-y_1\right)^2}&space;=&space;\sqrt{\left(x-x_2\right)^2+\left(y-y_2\right)^2}&space;=&space;\sqrt{\left(x-x_3\right)^2+\left(y-y_3\right)^2} "\sqrt{\left(x-x_1\right)^2+\left(y-y_1\right)^2} = \sqrt{\left(x-x_2\right)^2+\left(y-y_2\right)^2} = \sqrt{\left(x-x_3\right)^2+\left(y-y_3\right)^2}")
Incentre
The centre of the circle, which touches the sides of a triangle, is called its incentre.
Incentre is the point of intersection of internal angle bisectors of triangle.
If A (x1, y1) B (x2, y2) and C (x3, y3) are the vertices of a triangle ABC such that BC = a, CA = b and AB = c, then coordinates of its incentre I are
 "\left(\frac{ax_1+bx_2+cx_3}{a+b+c},\frac{ay_1+by_2+cy_3}{a+b+c}\right)")
Section formulae
- Let A (x1, y1) B (x2, y2) be two points on the cartesian plane.
- Let point P (x, y) divides the line AB in the ratio of m: n internally.
If P divides AB externally, then 
If P is the mid-point of AB, then 
Basic Points Related to Straight Lines
1. General form of equation of straight line is ax + by + c = 0. Where, a, b and c are real constants and x and y are two unknowns.
2. The equation of a line having slope m and intersects at c on x-axis is y = mx + c.
3. Slope (gradient) of a line ax + by + c = 0, by = - ax – c
Comparing with y = mx + c, where m is slope, therefore m = tan θ = 
Slope of the line is always measured in anti-clockwise direction.
4. Point slope form A line in terms of coordinates of any two points on it, if (x1, y1) and (x2, y2) are coordinates of any two points on a line, then its slope is
5. Two-point form a line the equation of a line passing through the points A (x1, y1) and B (x2, y2) is 
6. Condition of parallel lines
If the slopes of two lines i.e., m1 and m2 are equal then lines are parallel.
Equation of line parallel to ax + by + c = 0 is ax + by + q =
7. Condition of perpendicular lines
If the multiplication of slopes of two lines i.e., m1 and m2 is equal to -1 then lines are perpendicular.
m1 x m2 = -1
Equation of line perpendicular to ax + by + c = 0 is bx - ay + q =0
8. Angle between the two lines  "tan \theta = \pm \left(\frac{m_2-m_1}{1+m_1m_2}\right)")
9. Intercept form Equation of line L intersects at a and b on x and y-axes, respectively is 
10. Condition of concurrency of three lines:
Let the equation of three lines are a1x + b1y + c1 = 0,
a2x + b2y + c2 = 0, and a3x + b3y + c3 = 0.
Then, three lines will be concurrent, if
Distance of a point from the line:
Let ax + by + c = 0 be any equation of line and P (x1, y1) be any point in space. Then the perpendicular Distance(d) of a point P from a line is given by
12. The length of the perpendicular from the origin to the line ax + by + c = 0, is 
13. Area of triangle by straight line ax + by + c = 0 where a ≠ 0 and b ≠ 0 with coordinate axes is 
14. Distance between parallel lines ax + by + c = 0 and ax + by + d = 0 is equal to 
15. Area of trapezium, between two parallel lines and axes,
Area of trapezium ABCD = Area of OCD
 "Area of OAB =\frac{1}{2} \left|\frac{d^2}{ab}\right| - \frac{1}{2}\left|\frac{c^2}{ab}\right| = \frac{1}{2}(\left|\frac{d^2}{ab}\right| - \left|\frac{c^2}{ab}\right|)")
Examples:
Find the area of triangle ABC, whose vertices are A (8, - 4), B (3, 6) and C (- 2, 4).
Solution: Here, A (8, - 4) so, x1 = 8, y1 = - 4
B (3, 6) so, x2 = 3, y2 = 6
C (-2, 4) so, x3 = -2, y3 = 4
Therefore, area of triangle ABC 
(x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2))
(8(6 – 4) + 3(4 – (- 4)) + (-2) (-4-6))
(16 + 24 + 20)
= 30 sq units
If A (-2,1), B (2, 3) and C (-2, -4) are three points, then find the angle between AB and BC.
Solution: Let m1 and m2 be the slopes of line AB and BC, respectively.
}&space;=&space;\frac{2}{4}&space;=&space;\frac{1}{2}&space;and&space;m2&space;=&space;\frac{-4-3}{-2-2}&space;=&space;\frac{-7}{-4}&space;=&space;\frac{7}{4} "So, m1 = \frac{3-1}{2-(-2)} = \frac{2}{4} = \frac{1}{2} and m2 = \frac{-4-3}{-2-2} = \frac{-7}{-4} = \frac{7}{4}")
Let θ be the angle between AB and BC
} "\theta = {tan}^{-1}{\left(\frac{2}{3}\right)}")
3. In what ratio, the line made by joining the points A (- 4, - 3) and B (5,2) intersects x-axis?
Solution: We know that y-coordinate is zero on x-axis,
Given, y1 = - 3, y2 = 2
Therefore, 
+n(-3)}{m\&space;+n} "0 = \frac{m(2)+n(-3)}{m\ +n}")
2m – 3n = 0
4. Coordinates of a point is (0, 1) and ordinate of another point is - 3. If distance between both the points is 5, then abscissa of second point is
Solution: Let abscissa be x.
So, (x – 0)2 + (-3 -1)2 = 52
x2 + 16 = 25
x2 = 9
5. Do the points (4, 3), (- 4, - 6) and (7, 9) form a triangle? If yes, then find the longest side of the triangle
Solution: Let P (4, 3), Q (-4, -6) and R (7, 9) are given points.
Since, the sum of 12.04 and 6.7 is greater than 18.6.
So, it will form a triangle, whose longest side is 18.6
