1. ©G Dear 2010 – Not to be sold/Free to use
Mathematic Extension 1
(Preliminary)
Introduction
to Calculus
Gradient
Stage 6 - Year 11
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©G Dear 2010 – Not to be sold/Free to use

2. Dividing a Line in a Ratio
The coordinates of the point P(x,y) dividing
the line in the ratio m:n are:
f(x) x
B(x2,y2) ●
mx2 + nx1
m + n n
x = m
y =
my2 + ny1
m + n
P(x,y) ● ●
A(x1,y1) x1 x x2
End of
Slide

3. Dividing a Line in a Ratio
The coordinates of the point P(x,y) dividing
the line in the ratio m:n are:
f(x) x ●
A(x1,y1)
B(x2,y2)
P(x,y) ● x1 x2 m n
P(x,y) divides AB in ratio m:n.
Then = AP PB m n
AD = x – x1
DC = x2 – x
PD||BC
  = AP PB AD DC
(The intercepts
have equal ratios) D
 = m n
x-x1
x2-x
mx2 + nx1
(m + n)
m(x2-x) = n(x-x1)
x =
mx2 + nx1
m + n
mx2 - mx = nx - nx1
x =
Similarly for y.
mx2 + nx1 = mx + nx
y =
my2 + ny1
m + n
mx2 + nx1 = x(m + n)
End of
Slide