COORDINATE GEOMETRY
A
B
P
P divides seg AB internally in the ratio AP : PB.
Point P is called ‘Point of internal division’.
(x1, y1)
(x2, y2)
(x, y)
AP : PB
=
m1 : m2
SECTION FORMULA FOR INTERNAL DIVISION
=
m1
x2
+
+
m2
x1
m1
m2
x
=
m1
y2
+
+
m2
y1
m1
m2
y
Consider seg AB
Let A (x1, y1)
Let B (x2, y2)
Consider point P on seg AB such that A-P-B
Let P (x, y)
Now, let us understand the formula for finding coordinates of point ‘P’
Let point P divides seg AB in the ratio m : n
Section formula for internal division is used to find coordinates of point of internal division
A
B
P
(x1, y1)
(x2, y2)
(x, y)
AP : PB
=
m1 : m2
SECTION FORMULA FOR INTERNAL DIVISION
=
m1
x2
+
+
m2
x1
m1
m2
x
=
m1
y2
+
+
m2
y1
m1
m2
y
Let us understand, how to remember the formula
∴ P (1, 0)
x =
x
=
2
(4)
+
1
(–5)
2
+
1
=
8
–
5
3
=
3
3
=
2
(–4)
+
1
(8)
=
–8
+
8
3
By section formula for internal division,
1
=
m1
x2
+
m2
x1
m1 + m2
y =
m1
y2
+
m2
y1
m1 + m2
Sol.
y
∴
∴
Q. Find the coordinates of the point P which divides line segment
QR internally in the ratio m1 : m2.
1
=
0
2
+
1
Let the co-ordinates of Q be (x1, y1)
We have co-ordinates of two points and the ratio.
Let us substitute the values.
Let the co-ordinates of R be (x2, y2)
Which formula is used to find co-ordinates of P?
Section formula for Internal Division.
,
+
m1x2
m2x1
+
m2
m1
x
=
+
m1y2
m2y1
+
m2
m1
y
=
(i) Q (–5, 8) and R (4, –4) and m1 : m2 = 2:1
x1 = –5,
x2 = 4,
y2 = –4
y1 = 8