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Analytic geometry A level
shahbaz ahmed
August 2024

1

Introduction

Criteria for mutually perpendicular lines

Reference to the figure in triangle ABC
γ+β =α
γ =α−β
tan γ = tan(α − β)

1

tan γ =

tan α − tan β
1 + tan α tan β

(1)

Let m1 denotes the slope of the line l1
...

m1 = tan α
Similarly if m2 denotes the slope of the line l2 ,then
...

tan 90◦ = ∞
From equation (2)

∞=

m1 − m2
1 + m2 m2

⇐⇒
2

(3)

1 + m1 m2 = 0
m1 m2 = −1

2

Mid point formula

If A(x1 , x2 ) and B(y1 , y2 ) are end points of a line segment AB,then mid
point P is given by:
2 y1 +y2
P ( x1 +x
2 , 2 )

3

4

3

Equation of a line passing through two points A(x1 , y1 )
and B(x2 , y2 )

5

Reference to the figure
...

tan α = m =

y2 −y1
x2 −x1

is the slope of the line
...

Solution
...
Then the
perpendicular bisector of AB passes through the mid point
5+8
13
C( 3+9
2 , 2 ) = C(6, 2 ) = C(x1 , y1 )
...
=⇒

6

m1 m = −1
Then
1
2m

= −1

m = −2
Equation of the line passing through the point C(6, 13
2 ) = C(x1 , y1 )
with m = −2
y−

13
= −2(x − 6)
2

2y − 13 = −4(x − 6)
2y − 13 = −4x + 24
4x + 2y − 13 − 24 = 0
4x + 2y − 37 = 0

7

8



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