Page 1
1. Find the coordinates of the point which divides the join of 1,7 4, 3 . A and B in the ratio
2 : 3
Sol:
The end points of AB are 1,7 4, 3 . A and B
Therefore,
1 1 2 2
1 , 7 4, 3 x y and x y
Also, 2 3 m and n
Let the required point be , . P x y
By section formula, we get
2 1 2 1
,
2 4 3 1 2 3 3 7
,
2 3 2 3
8 3 6 21
,
5 5
5 15
,
5 5
mx nx my ny
x y
m n m n
x y
x y
x y
Therefore, 1 3 x and y
Hence, the coordinates of the required point are 1,3 .
2. Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the
ratio 7 : 2
Sol:
The end points of AB are 5,11 4, 7 . A and B
Therefore,
1 1 2 2
5, 11 4, 7 x y and x y
Also, 7 2 m and n
Let the required point be , . P x y
By section formula, we get
2 1 2 1
,
7 4 2 5 7 7 2 11
,
7 2 7 2
28 10 49 22
,
9 9
mx nx my ny
x y
m n m n
x y
x y
Page 2
1. Find the coordinates of the point which divides the join of 1,7 4, 3 . A and B in the ratio
2 : 3
Sol:
The end points of AB are 1,7 4, 3 . A and B
Therefore,
1 1 2 2
1 , 7 4, 3 x y and x y
Also, 2 3 m and n
Let the required point be , . P x y
By section formula, we get
2 1 2 1
,
2 4 3 1 2 3 3 7
,
2 3 2 3
8 3 6 21
,
5 5
5 15
,
5 5
mx nx my ny
x y
m n m n
x y
x y
x y
Therefore, 1 3 x and y
Hence, the coordinates of the required point are 1,3 .
2. Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the
ratio 7 : 2
Sol:
The end points of AB are 5,11 4, 7 . A and B
Therefore,
1 1 2 2
5, 11 4, 7 x y and x y
Also, 7 2 m and n
Let the required point be , . P x y
By section formula, we get
2 1 2 1
,
7 4 2 5 7 7 2 11
,
7 2 7 2
28 10 49 22
,
9 9
mx nx my ny
x y
m n m n
x y
x y
18 27
,
9 9
x y
Therefore, 2 3 x and y
Hence, the required point are 2, 3 . P
3. If the coordinates of points A and B are (-2, -2) and (2, -4) respectively. Find the
coordinates of the point P such that AP=
3
7
AB, where P lies on the segment AB.
Sol:
The coordinates of the points A and Bare 2, 2 and 2, 4 respectively, where
3
7
AP AB and P lies on the line segment AB. So
7 3
3 7
AP BP AB
AP
AP BP AP AB
7
3
3
4
AP
BP AP
AP
BP
Let (x, y) be the coordinates of P which divides AB in the ratio 3 : 4 internally Then
3 2 4 2
6 8 2
3 4 7 7
3 4 4 2
12 8 20
3 4 7 7
x
y
Hence, the coordinates of point Pare
2 20
, .
7 7
4. Point A lies on the line segment PQ joining P(6, -6) and Q(-4, -1) in such a way that
2
5
PA
PQ
. If that point A also lies on the line 3x + k( y + 1 ) = 0, find the value of k.
Sol:
Let the coordinates of A be , . x y Here
2
.
5
PA
PQ
So,
PA AQ PQ
5 2
2 5
PA
PA AQ PA PQ
Page 3
1. Find the coordinates of the point which divides the join of 1,7 4, 3 . A and B in the ratio
2 : 3
Sol:
The end points of AB are 1,7 4, 3 . A and B
Therefore,
1 1 2 2
1 , 7 4, 3 x y and x y
Also, 2 3 m and n
Let the required point be , . P x y
By section formula, we get
2 1 2 1
,
2 4 3 1 2 3 3 7
,
2 3 2 3
8 3 6 21
,
5 5
5 15
,
5 5
mx nx my ny
x y
m n m n
x y
x y
x y
Therefore, 1 3 x and y
Hence, the coordinates of the required point are 1,3 .
2. Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the
ratio 7 : 2
Sol:
The end points of AB are 5,11 4, 7 . A and B
Therefore,
1 1 2 2
5, 11 4, 7 x y and x y
Also, 7 2 m and n
Let the required point be , . P x y
By section formula, we get
2 1 2 1
,
7 4 2 5 7 7 2 11
,
7 2 7 2
28 10 49 22
,
9 9
mx nx my ny
x y
m n m n
x y
x y
18 27
,
9 9
x y
Therefore, 2 3 x and y
Hence, the required point are 2, 3 . P
3. If the coordinates of points A and B are (-2, -2) and (2, -4) respectively. Find the
coordinates of the point P such that AP=
3
7
AB, where P lies on the segment AB.
Sol:
The coordinates of the points A and Bare 2, 2 and 2, 4 respectively, where
3
7
AP AB and P lies on the line segment AB. So
7 3
3 7
AP BP AB
AP
AP BP AP AB
7
3
3
4
AP
BP AP
AP
BP
Let (x, y) be the coordinates of P which divides AB in the ratio 3 : 4 internally Then
3 2 4 2
6 8 2
3 4 7 7
3 4 4 2
12 8 20
3 4 7 7
x
y
Hence, the coordinates of point Pare
2 20
, .
7 7
4. Point A lies on the line segment PQ joining P(6, -6) and Q(-4, -1) in such a way that
2
5
PA
PQ
. If that point A also lies on the line 3x + k( y + 1 ) = 0, find the value of k.
Sol:
Let the coordinates of A be , . x y Here
2
.
5
PA
PQ
So,
PA AQ PQ
5 2
2 5
PA
PA AQ PA PQ
5
2
PA
AQ PA
3
2
AQ
PA
2
3
PA
AQ
Let (x, y) be the coordinates of A, which dives PQ in the ratio 2 : 3 internally Then using
section formula, we get
2 4 3 6
8 18 10
2
2 3 5 5
2 1 3 6
2 18 20
4
2 3 5 5
x
y
Now, the point 2, 4 lies on the line3 1 0, x k y therefore
3 2 4 1 0
3 6
6
2
3
k
k
k
Hence, 2. k
5. Points P, Q, R and S divide the line segment joining the points A(1,2) and B(6,7) in five
equal parts. Find the coordinates of the points P,Q and R
Sol:
Since, the points P, Q, R and S divide the line segment joining the points
1,2 6,7 A and B in five equal parts, so
AP PQ QR R SB
Here, point P divides AB in the ratio of 1 : 4 internally So using section formula, we get
Coordinates of
1 6 4 1 1 7 4 2
,
1 4 1 4
P
6 4 7 8
, 2,3
5 5
The point Q divides AB in the ratio of 2 : 3 internally. So using section formula, we get
Coordinates of
2 6 3 1 2 7 3 2
,
2 3 2 3
Q
12 3 14 6
, 3,4
5 5
The point R divides AB in the ratio of 3 : 2 internally So using section formula, we get
Page 4
1. Find the coordinates of the point which divides the join of 1,7 4, 3 . A and B in the ratio
2 : 3
Sol:
The end points of AB are 1,7 4, 3 . A and B
Therefore,
1 1 2 2
1 , 7 4, 3 x y and x y
Also, 2 3 m and n
Let the required point be , . P x y
By section formula, we get
2 1 2 1
,
2 4 3 1 2 3 3 7
,
2 3 2 3
8 3 6 21
,
5 5
5 15
,
5 5
mx nx my ny
x y
m n m n
x y
x y
x y
Therefore, 1 3 x and y
Hence, the coordinates of the required point are 1,3 .
2. Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the
ratio 7 : 2
Sol:
The end points of AB are 5,11 4, 7 . A and B
Therefore,
1 1 2 2
5, 11 4, 7 x y and x y
Also, 7 2 m and n
Let the required point be , . P x y
By section formula, we get
2 1 2 1
,
7 4 2 5 7 7 2 11
,
7 2 7 2
28 10 49 22
,
9 9
mx nx my ny
x y
m n m n
x y
x y
18 27
,
9 9
x y
Therefore, 2 3 x and y
Hence, the required point are 2, 3 . P
3. If the coordinates of points A and B are (-2, -2) and (2, -4) respectively. Find the
coordinates of the point P such that AP=
3
7
AB, where P lies on the segment AB.
Sol:
The coordinates of the points A and Bare 2, 2 and 2, 4 respectively, where
3
7
AP AB and P lies on the line segment AB. So
7 3
3 7
AP BP AB
AP
AP BP AP AB
7
3
3
4
AP
BP AP
AP
BP
Let (x, y) be the coordinates of P which divides AB in the ratio 3 : 4 internally Then
3 2 4 2
6 8 2
3 4 7 7
3 4 4 2
12 8 20
3 4 7 7
x
y
Hence, the coordinates of point Pare
2 20
, .
7 7
4. Point A lies on the line segment PQ joining P(6, -6) and Q(-4, -1) in such a way that
2
5
PA
PQ
. If that point A also lies on the line 3x + k( y + 1 ) = 0, find the value of k.
Sol:
Let the coordinates of A be , . x y Here
2
.
5
PA
PQ
So,
PA AQ PQ
5 2
2 5
PA
PA AQ PA PQ
5
2
PA
AQ PA
3
2
AQ
PA
2
3
PA
AQ
Let (x, y) be the coordinates of A, which dives PQ in the ratio 2 : 3 internally Then using
section formula, we get
2 4 3 6
8 18 10
2
2 3 5 5
2 1 3 6
2 18 20
4
2 3 5 5
x
y
Now, the point 2, 4 lies on the line3 1 0, x k y therefore
3 2 4 1 0
3 6
6
2
3
k
k
k
Hence, 2. k
5. Points P, Q, R and S divide the line segment joining the points A(1,2) and B(6,7) in five
equal parts. Find the coordinates of the points P,Q and R
Sol:
Since, the points P, Q, R and S divide the line segment joining the points
1,2 6,7 A and B in five equal parts, so
AP PQ QR R SB
Here, point P divides AB in the ratio of 1 : 4 internally So using section formula, we get
Coordinates of
1 6 4 1 1 7 4 2
,
1 4 1 4
P
6 4 7 8
, 2,3
5 5
The point Q divides AB in the ratio of 2 : 3 internally. So using section formula, we get
Coordinates of
2 6 3 1 2 7 3 2
,
2 3 2 3
Q
12 3 14 6
, 3,4
5 5
The point R divides AB in the ratio of 3 : 2 internally So using section formula, we get
Coordinates of
3 6 2 1 3 7 2 2
,
3 2 3 2
R
18 2 21 4
, 4,5
5 5
Hence, the coordinates of the points P, Q and R are 2,3 , 3,4 and 4,5 respectively
6. Points P, Q, and R in that order are dividing line segment joining A (1,6) and B(5, -2) in
four equal parts. Find the coordinates of P, Q and R.
Sol:
The given points are 1,6 A and 5, 2 . B
Then, , P x y is a point that devices the line AB in the ratio 1: 3
By the section formula:
2 1 2 1
,
1 2 3 6 1 5 3 1
,
1 3 1 3
5 3 2 18
,
4 4
8 16
,
4 4
2 4
mx nx my ny
x y
m n m n
x y
x y
x y
x and y
Therefore, the coordinates of point P are 2,4
Let Q be the mid-point of AB
Then, , Q x y
1 2 1 2
,
2 2
6 2
1 5
,
2 2
6 4
,
2 2
3, 2
x x y y
x y
x y
x y
x y
Therefore, the coordinates of Q are 3,2
Let , R x y be a point that divides AB in the ratio 3 :1
Then, by the section formula:
Page 5
1. Find the coordinates of the point which divides the join of 1,7 4, 3 . A and B in the ratio
2 : 3
Sol:
The end points of AB are 1,7 4, 3 . A and B
Therefore,
1 1 2 2
1 , 7 4, 3 x y and x y
Also, 2 3 m and n
Let the required point be , . P x y
By section formula, we get
2 1 2 1
,
2 4 3 1 2 3 3 7
,
2 3 2 3
8 3 6 21
,
5 5
5 15
,
5 5
mx nx my ny
x y
m n m n
x y
x y
x y
Therefore, 1 3 x and y
Hence, the coordinates of the required point are 1,3 .
2. Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the
ratio 7 : 2
Sol:
The end points of AB are 5,11 4, 7 . A and B
Therefore,
1 1 2 2
5, 11 4, 7 x y and x y
Also, 7 2 m and n
Let the required point be , . P x y
By section formula, we get
2 1 2 1
,
7 4 2 5 7 7 2 11
,
7 2 7 2
28 10 49 22
,
9 9
mx nx my ny
x y
m n m n
x y
x y
18 27
,
9 9
x y
Therefore, 2 3 x and y
Hence, the required point are 2, 3 . P
3. If the coordinates of points A and B are (-2, -2) and (2, -4) respectively. Find the
coordinates of the point P such that AP=
3
7
AB, where P lies on the segment AB.
Sol:
The coordinates of the points A and Bare 2, 2 and 2, 4 respectively, where
3
7
AP AB and P lies on the line segment AB. So
7 3
3 7
AP BP AB
AP
AP BP AP AB
7
3
3
4
AP
BP AP
AP
BP
Let (x, y) be the coordinates of P which divides AB in the ratio 3 : 4 internally Then
3 2 4 2
6 8 2
3 4 7 7
3 4 4 2
12 8 20
3 4 7 7
x
y
Hence, the coordinates of point Pare
2 20
, .
7 7
4. Point A lies on the line segment PQ joining P(6, -6) and Q(-4, -1) in such a way that
2
5
PA
PQ
. If that point A also lies on the line 3x + k( y + 1 ) = 0, find the value of k.
Sol:
Let the coordinates of A be , . x y Here
2
.
5
PA
PQ
So,
PA AQ PQ
5 2
2 5
PA
PA AQ PA PQ
5
2
PA
AQ PA
3
2
AQ
PA
2
3
PA
AQ
Let (x, y) be the coordinates of A, which dives PQ in the ratio 2 : 3 internally Then using
section formula, we get
2 4 3 6
8 18 10
2
2 3 5 5
2 1 3 6
2 18 20
4
2 3 5 5
x
y
Now, the point 2, 4 lies on the line3 1 0, x k y therefore
3 2 4 1 0
3 6
6
2
3
k
k
k
Hence, 2. k
5. Points P, Q, R and S divide the line segment joining the points A(1,2) and B(6,7) in five
equal parts. Find the coordinates of the points P,Q and R
Sol:
Since, the points P, Q, R and S divide the line segment joining the points
1,2 6,7 A and B in five equal parts, so
AP PQ QR R SB
Here, point P divides AB in the ratio of 1 : 4 internally So using section formula, we get
Coordinates of
1 6 4 1 1 7 4 2
,
1 4 1 4
P
6 4 7 8
, 2,3
5 5
The point Q divides AB in the ratio of 2 : 3 internally. So using section formula, we get
Coordinates of
2 6 3 1 2 7 3 2
,
2 3 2 3
Q
12 3 14 6
, 3,4
5 5
The point R divides AB in the ratio of 3 : 2 internally So using section formula, we get
Coordinates of
3 6 2 1 3 7 2 2
,
3 2 3 2
R
18 2 21 4
, 4,5
5 5
Hence, the coordinates of the points P, Q and R are 2,3 , 3,4 and 4,5 respectively
6. Points P, Q, and R in that order are dividing line segment joining A (1,6) and B(5, -2) in
four equal parts. Find the coordinates of P, Q and R.
Sol:
The given points are 1,6 A and 5, 2 . B
Then, , P x y is a point that devices the line AB in the ratio 1: 3
By the section formula:
2 1 2 1
,
1 2 3 6 1 5 3 1
,
1 3 1 3
5 3 2 18
,
4 4
8 16
,
4 4
2 4
mx nx my ny
x y
m n m n
x y
x y
x y
x and y
Therefore, the coordinates of point P are 2,4
Let Q be the mid-point of AB
Then, , Q x y
1 2 1 2
,
2 2
6 2
1 5
,
2 2
6 4
,
2 2
3, 2
x x y y
x y
x y
x y
x y
Therefore, the coordinates of Q are 3,2
Let , R x y be a point that divides AB in the ratio 3 :1
Then, by the section formula:
2 1 2 1
,
mx nx my ny
x y
m n m n
3 2 1 6 3 5 1 1
,
3 1 3 1
15 1 6 6
,
4 4
16 0
,
4 4
4 0
x y
x y
x y
x and y
Therefore, the coordinates of R are 4,0 .
Hence, the coordinates of point P, Q and R are 2,4 , 3,2 4,0 and respectively.
7. The line segment joining the points A(3,-4) and B(1,2) is trisected at the points P(p, -2) and
5
,
3
Q q . Find the values of p and q.
Sol:
Let P and Q be the points of trisection of AB.
Then, P divides AB in the radio 1:2
So, the coordinates of P are
2 1 2 1
,
1 1 2 3 1 2 2 4
,
1 2 1 2
1 6 2 8
,
3 3
7 6
,
3 3
7
, 2
3
mx nx my ny
x y
m n m n
x y
x y
x y
x y
Hence, the coordinates of P are
7
, 2
3
But , 2 p are the coordinates of P.
So,
7
3
p
Also, Q divides the line AB in the ratio 2:1
So, the coordinates of Q are
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