Coordinate Geometry

1. Coordinate Geometry
Concept Session

2. The Cartesian plane
2

3. Distance between 2 Points P(x1, y1) and Q(x2,
y2)
Distance PQ :
𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2
Distance from origin of point P :
𝑥1
2 + 𝑦1
2
3

4. Distance between 2 Points - Problems
Let the vertices of a triangle ABC be
(−8, 8), (2, −8) and (−2, 2) then the triangle is
A. Right angled
B. Equilateral
C. Isosceles
D. Scalene
E. Cannot be determined
4

5. The midpoint of an interval
The midpoint of an interval with endpoints
𝑃(𝑥1, 𝑦1) and 𝑄(𝑥2, 𝑦2) is
𝑥1 + 𝑥2
2
,
𝑦1 + 𝑦2
2
5

6. The midpoint of an interval - Problem
If C(3, 5) is the midpoint of line interval AB
and A has coordinates (–2, 1), find the
coordinates of B.
6
Answer : (8, 9)

7. Section formula – Internal
division
Given two end points of line segment A(x1, y1) and B
(x2, y2) you can determine the coordinates of the point
P(x, y) that divides the given line segment in the ratio
m : n internally using Section Formula
𝑥 =
𝑚𝑥2+𝑛𝑥1
𝑚+𝑛
𝑦 =
𝑚𝑦2+𝑛𝑦1
𝑚+𝑛
7

8. Section formula – External
division
Given two end points of line segment A(x1, y1) and B
(x2, y2) you can determine the coordinates of the point
P(x, y) that divides the given line segment in the ratio
m : n externally using Section Formula given by
𝑥 =
𝑚𝑥2−𝑛𝑥1
𝑚−𝑛
𝑦 =
𝑚𝑦2−𝑛𝑦1
𝑚−𝑛
8

9. Area of a Triangle
The area of a triangle ABC whose vertices are
A(x1, y1), B (x2, y2) and C(x3, y3) is denoted by
1
2
| 𝑥1 𝑦2 − 𝑥2 𝑦1 + 𝑥2 𝑦3 − 𝑥3 𝑦2
+ 𝑥3 𝑦1 − 𝑥1 𝑦3 |
9

10. Centroid of a triangle
Centroid – meeting point of the three medians.
The centroid(G) of a triangle ABC whose vertices are
A(𝑥1, 𝑦1), B(𝑥2, 𝑦2) and C(𝑥3, 𝑦3) is denoted by
𝐺 =
𝑥1 + 𝑥2 + 𝑥3
3
,
𝑦1 + 𝑦2 + 𝑦3
3
10

11. Straight Line
▪Slope of a Line
▪Equations of a straight line
–General form of Equation
–Slope-intercept form
–Point slope form
–Two point form
–Intercept form
11

12. Slope of a line ( Gradient)
The slope, represented by the letter m, measures
the inclination or steepness of the line.
The slope is always measured anti-clock wise
𝑆𝑙𝑜𝑝𝑒 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
=
𝑅𝑖𝑠𝑒
𝑅𝑢𝑛
𝑆𝑙𝑜𝑝𝑒 =
∆𝑦
∆𝑥
=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
12

13. Slope of a line ( Gradient) - values
Positive Slope – On moving from
left to right, the line rises
Negative Slope - On moving from left
to right, the line dips
13

14. Slope of a line ( Gradient) - values
Zero Slope – Parallel to x axis and
hence “No rise”
Undefined Slope – Parallel to y axis
and hence “ infinite rise”
14

15. Slope of a special line
Parallel Lines  equal slope
𝑚1 = 𝑚2
Perpendicular lines  negative
inverses
𝑚1 ∗ 𝑚2 = −1
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16. Problems
16
Quantity A Quantity B
Sole of a line
perpendicular to
Line 1
Slope of a line
perpendicular to
Line 2

17. Equation of a straight line – General Form
𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0
Slope of the line =
−𝑎
𝑏
Y intercept =
𝑐
𝑏
17

18. Equation of a straight line – Slope intercept Form
Equation of a straight line whose slope
is 𝑚 and which cuts of a y-intercept of 𝐶
units is given by
𝑦 = 𝑚𝑥 + 𝐶
18

19. Equation of a straight line – Problems
Find the equation of a line that
intersects the Y axis at the point
(0,3) and has a slope of
5
3
Answer : 3𝑦 − 5𝑥 = 9
19

20. Equation of a straight line – Point-slope Form
Equation of a straight line passing
through a point (𝑥1, 𝑦1) and having a
slope m is given by
𝑦 − 𝑦1 = 𝑚 ∗ (𝑥 − 𝑥1)
20

21. Equation of a straight line – Problems
Find an equation of the line that passes through
(4, 6) and is parallel to the line whose equation is
𝑦 =
2
3
𝑥 + 5.
a) 2𝑦 = 3𝑥 + 10
b) 3𝑦 = 3𝑥 + 10
c) 3𝑦 = 2𝑥 + 10
d) 4𝑦 = 3𝑥 + 15
e) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒
Answer: Option
3
21

22. Equation of a straight line – Problems
Find an equation of the line that passes through
(4, 6) and is perpendicular to the line whose
equation is 𝑦 =
2
3
𝑥 + 5.
1. 2𝑦 = −3𝑥 + 10
2. 3𝑦 = −3𝑥 + 24
3. 3𝑦 = 2𝑥 + 10
4. 4𝑦 = 3𝑥 + 13
5. 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒
Answer: 2𝑦 = −3𝑥 + 24
22

23. Equation of a straight line – Two-point Form
Equation of a straight line passing
through two points (𝑥1, 𝑦1) and (𝑥2, 𝑦2)
is given by
𝑦 − 𝑦1 =
(𝑦2−𝑦1)
(𝑥2−𝑥1)
∗ (𝑥 − 𝑥1)
23

24. Equation of a straight line – Intercept Form
Equation of a straight
line which cuts off x-
intercept (a) and y-
intercept (b) is
𝑥
𝑎
+
𝑦
𝑏
= 1
24

25. Transformations – Vertical and Horizontal Shifts
25
Vertical Shift
To Graph Shift the Graph of ƒ(x)
ƒ(𝑥) + 𝑐 c units upward
ƒ 𝑥 − 𝑐 c units upward
Horizontal Shift
To Graph Shift the Graph of ƒ(x)
ƒ(𝑥 + 𝑐) c units to the left
ƒ 𝑥 − 𝑐 c units to the right

26. Transformations – Reflections
26
Reflection along coordinate axis
To Graph Shift the Graph of ƒ(x)
−ƒ(𝑥) Reflect along x axis
ƒ −𝑥 Reflect along y axis

27. Problems
27
Which of the following
could be the equation of
the figure given ?
A. 𝑦 = 𝑥2
B. 𝑦 = 𝑥2
+ 9
C. 𝑦 = 10𝑥 + 6
D. 𝑦 = 𝑥2
− 4
E. 𝑦 = 𝑥3
+ 4

28. Problems
28
Which of the following
could be the equation of
the figure given ?
A. 𝑦 = 𝑥2
B. 𝑦 = −𝑥2
+ 9
C. 𝑦 = −𝑥 + 6
D. 𝑦 = −(𝑥 + 2)2
− 4
E. 𝑦 = − 𝑥 + 2 2
+ 2

29. Sin, cos and tan values
𝟎° 𝟑𝟎° 𝟒𝟓° 𝟔𝟎° 𝟗𝟎°
Sin 0
1
2
1
2
3
2
1
Cos
1 3
2
1
2
1
2
0
Tan
0 1
3
1
3 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
29

30. Conditions of Collinearity of three Points
1. Area of triangle ABC is zero.
2. Slope of AB = Slope of BC = Slope of AC
3. Distance between A and B + Distance between
B and C = Distance between A and C
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31. Practice Problems
If 𝐴(1,2), 𝐵(4,3), 𝐶(6,6) and 𝐷(𝑥, 𝑦) are the
four vertices of a parallelogram taken in an
order, then the value of 𝑥 + 𝑦 is
A. 5
B. 7
C. 8
D. 10
E. None of these
Answer : option 3
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