Coordinate Geometry is the branch of mathematics related to the study of algebraic equations through the geometrical interpretation. In this branch, we plot the graphs over the three dimensions to show points, lines, curves, planes, surfaces, etc.
Quadrants
Polar coordinates system specifies a point by the distance (r) from a reference point (known as origin or pole) and angle (Ɵ) from an axis.
A point in a polar coordinate system is defined as (r, Ɵ).
Polar Coordinate System
Example 1: Convert the point P(5, 30^{o}) into cartesian form
Ans: Step 1: Firstly, find the value of x coordinate
We know,
x = rcosƟ
⇒x = 5 cos 30^{o}
Step 2: Secondly, find the value of y coordinate
y = rsinƟ
⇒ y = 5 sin 30^{o}
Therefore,
The distance between two points whose coordinates are A (x_{1}, y_{1})and B (x_{2}, y_{2}) will be
Distance FormulaNote:
 The distance of any point P(x, y) from the origin can be given as
 In a polar coordinate system distance between two points A(r_{1}, Ɵ_{1}) and B(r_{2}, Ɵ_{2}) is given as
 Distance= r_{1}^{2} + r_{2}^{2} – 2r_{1} r_{2} cos (Ɵ_{1} – Ɵ_{2})
Section formula tells us about the coordinates of the Point, which divides a line segment into the ratio m : n.
External Division of a line
Note:
 The coordinates of a point A(x, y) which divides the line segment PQ into two equal halves (i.e., m : n = 1 : 1) is given as:
 If a point A divides line PQ internally in the ratio m : n and the point B divides the line segment PQ externally in the ratio m : n, then
The slope of a line denotes the steepness of a line. Slope is the tangent of the angle made by straight line with the positive direction of Xaxis. It can also be defined as the change in Ycoordinate per unit change in Xcoordinate.
Slope of a Line
The slope of a line AB having coordinates A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is given as
Note: Slope of Xaxis is 0 and the slope of Yaxis is not defined.
Example 2: Find the slope of the line passing through the coordinates A(1, 9) and B(3, 5)?
Sol: We know,
Thus, slope of the given line will be 7.
There are many forms a straight line can be represented:
x cos ∅ + y sin ∅ = p
Note: The slope of a line can also be defined as the coefficient of x when the coefficient of y is 1.
Example 3: Find the slope of the equation 3y = 4x + 9
Sol: Step 1: Firstly, make the coefficient of y as 1
3y = 4x + 9Step 2: Now, the slope of equation is the coefficient of x i.e. slope (m) = 4/3
Position of point. A(x_{1}, y_{1}) and B(x_{2}, y_{2}) with respect to line ax + by + c = 0
If the value of ax_{1} + by_{1} + c and ax^{2} + by^{2} + c are of the same sign than the points lie on the same side of straightline otherwise on the different side of the line.
Note: For a point (x_{1}, y_{1}), if ax_{1} + by_{1} + c is equal to zero, then the point lies on the line ax + by + c = 0
The family of equation of lines parallel to line ax + by + c = 0 is given as ax + by + d = 0
When two lines are parallel to each other then, the angle between them is zero and slopes of them are equal, i.e., m_{1} = m_{2}
The family of equation of lines perpendicular to line ax + by + c = 0 is given as bx – ay + d = 0.
Example 4: If the line formed by the points (1, 4) and (3, k) is perpendicular to the line 3x – 4y = 5, then what will be the value of k?
Sol: Step 1: Find the slope of line 3x – 4y = 5
Slope is the coefficient of x when coefficient of y is 1.
So, slope for line will be m_{1} = 3/4
Step 2: Find the slope for line formed by points (1, 4) and (3, k)
Step 3: for two lines perpendicular to each other m_{1} * m_{2} = 1
The shortest distance from a point P (x_{1}, y_{1}) to the straight line having equation Ax + By + c = 0 will be the length of the perpendicular drawn from point P to the line Ax + By + C = 0.
The points are said to be collinear when all the points lie in a straight line.
The points A (x_{1}, y_{1}), B (x_{2}, y_{2}) and C(x_{3}, y_{3}) will be collinear to each other if slope of line segment AB is equal to the slope of line segment BC. Equating slope of both the lines we get a direct result:
The pair of lines is said to be concurrent to each other if all the lines intersect each other at the common point P, which is known as Point of concurrency.
Concurrent Lines
The lines a_{1}x + b_{1}y +c_{1} = 0, a_{2}x + b_{2}y +c_{2} = 0 , and a_{3}x + b_{3}y + c_{3} = 0 will be concurrent to each other if and only if
To find the projection of a point P (x_{1}, y_{1}) on a line Ax + By + C = 0 can be found using the following steps:
Step 1: First find the equation of line perpendicular to Ax + By + C = 0
The line perpendicular to Ax + By + C = 0 will be Bx – Ay + D = 0
Step 2: Now, put the coordinates of point P (x_{1}, y_{1}) in the line Bx – Ay + D = 0 as the points satisfy the equation to find the value of D.
⇒ Bx_{1} – Ay_{1} + D = 0
Step 3: Now, find the intersection point of line Ax + By + C = 0 and Bx – Ay + D = 0 to obtain the projection of a point on a line Ax + By + C = 0
Example 5: What will be the projection of (3, 1) on the line 5x – 3y = 4?
Sol:
Step 1: Equation of Line perpendicular to 5x – 3y = 4 will be
⇒ 3x – 5y = D
Step 2: Putting coordinates of (3, 1) in the above equation to find the value of D
⇒ D = 3 * 3 – 5 * 1 = 9 – 5 = 4
Step 3: Find the intersection point of 5x – 3y = 4 and 3x – 5y = 4
Solving above equation for x and y we get
x = 1/2 and y = 1/2
Thus, projection of (3, 1) on the line 5x – 3y = 4 is
Reflection of a point about X axis and Y axis respectively
The coordinates of a centroid of a triangle ABC having vertices as A (x_{1}, y_{1}), B (x_{2}, y_{2}) and C(x_{3}, y_{3}) is given as
Centroid Formula
Note: Centroid divides the line formed by joining Orthocentre and Circumcentre internally in the ratio 2 : 1.
The coordinates of an Incentre of a triangle ABC having vertices as A (x_{1}, y_{1}), B (x_{2}, y_{2}) and C (x_{3}, y_{3}) and sides a = BC, b = CA and c = AB is given as
Incentre of a Triangle
The area of a triangle ABC formed from the vertices A (x_{1}, y_{1}), B (x_{2}, y_{2}) and C (x_{3}, y_{3}) will be
Note:
 The vertices should be taken in either a clockwise direction or an anticlockwise direction.
 If the area of the triangle is zero, then all the vertices A, B, and C are collinear to each other, i.e., all the points lie on a straight line.
 The Area of triangle for line y = m_{1}x + c_{1}, y = m_{2}x + c_{2} and x = 0 is given as
The area of a polygon having n vertices in the form (x_{i}, y_{i}) where i = 1, 2, 3, ……, n can be calculated as
Example 6: Find the area of the quadrilateral having vertices (1, 4), (2, 6), (5, 4) and (4, 3)
Sol:
Step 1: write x coordinates of all the vertices in a column and write y coordinates of all the vertices in another column. Repeat the coordinates of first point in the last row to complete the calculation of area.Step 2: Now multiply the first row coordinate of x with secondrow coordinate of y and then from the obtained result subtract product of first row coordinate of y and second row coordinate of x. Do the step for all the coordinates and then add all the numbers thus obtained
⇒ (1*6 – 2*4) + (2*4 – 5*6) + (5*3 – 4*4) + (4*4 – 1*3)
⇒ 2 – 22 – 1 + 13
⇒  12
Thus, area of quadrilateral will be
1. What is the process for converting polar coordinates into cartesian coordinates? 
2. How is the slope of a straight line calculated in coordinate geometry? 
3. What is the equation of a straight line in coordinate geometry? 
4. How are the angles between two lines determined in coordinate geometry? 
5. How can the shortest distance between two parallel lines be found in coordinate geometry? 

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