Question 11
Question: Show by section formula, that the points (3,-2), (5,2) and (8,8) are collinear. [2 Mark]
Answer: Let A, B, C be the points whose co-ordinates are (3,-2), (5,2) and (8,8) respectively.
Any point on AC, dividing the segment in the ratio 
:1 is
If this point is B, then
8
+ 3 = 5
+ 5 and 8
- 2 = 2
+ 2
Solving both the equations, we get 
that is B divides AC in the ratio 2:3.
B lies on the line AC.
Hence, the three given points A, B and C are collinear.
Question 12
Question: If the points (-2,-1), (1, 0), (x, 3), (1, y) form a parallelogram, the find the value of x, y. [2 Mark]
Answer: 
The diagonals of a parallelogram bisect each other. That is,
The midpoint of AC = The midpoint of BD
That is 
(x, y) = (4, 2)
Question 13
Question: Find the co-ordinates of the point dividing the line segment joining the points: [Each 2 Mark]
(i) A(5, 7), B(2, 4) in the ratio 2:1.
(ii) A(-6, 6), B(-1, 6) in the ratio 2:3
Answer: (i) Let P(x, y) be the point which divides AB in the ratio 2:1.
Then co-ordinates of P are given by
m = 2 and n = 1,
Coordinates of P(x, y) = P(3, 5).
(ii) A(-6, 6), B(-1, 6) in the ratio 2:3
Let point P(x, y) divide the line joining points A(x1, y1), B(x2, y2) in the ratio 2:3
x1 = -1y2 = 6
m = 2, n = 3
Then P(x, y) is given by
x = -4 y = 6
Coordinates of P(-4, 6).
Question 14
Question: Find the coordinates of the mid points of the line segment joining the points: [Each 2 Mark]
(i) (8, -3), (-4, -7)
(ii) (4b, 2b), (b, 6b)
Answer: If A(x1, y1), B(x2, y2) is the end points of line segment, then mid point P(x, y) is given by the formula,
P(x, y) = (2, -5)
(ii) (4b, 2b), (b, 6b)
Let x1 = 4b y1 = 2b
x2 = b y2 = 6b
Question 15
Question: In what ratio does the point P(8, 4) divide the join of A(5, -2) and B(9, 6)? [2 Mark]
Answer: Let the point P(8, 4) divide the line joining Point A(5, -2) and B(9, 6) in the ratio m:n.
3n = m 6n = 2m
m:n = 3:1
P divides line AB in the ratio 3:1.
Question 16
Question: In what ratio does the point C(-3, 0) divide the join of A(-7, 2) and B(-1, -1)? [2 Mark]
Answer: Let the point C(-3, 0) divide the line joining A(-7, 2) and (-1, -1) in the ratio m:n.
Then according to section formula,
Question 17
Question: In what ratio is the line segment joining the points (5, -4) and (-3, 2) divided by X- axis? [2 Mark]
Answer: 
Let the point C on X-axis divide the line joining AB in the ratio m:n
Hence its coordinates will be (x, 0), coordinates of C are given by,
0 = 2m - 4n
m:n = 2:1
Question 18
Question: In what ratio is the line segment joining the points (-2, -3) and (5, 4) divided by y - axis? Also write the coordinates of the points of intersection on y - axis. [2 Mark]
Answer: Let the Point on y - axis C(0, y) divide the line joining A(-2, -3) and B(5, 4) in ratio m:n.
By section formula,
Hence y - axis divides the line segment in the ratio 2:5.
(as multiplication is commutative)
(0, 1) is the point on y - axis.
Question 19
Question: In what ratio is the line segment joining the points (9, 2) and (-3, -2) divided by y - axis? Also write the coordinates of the point of intersection. [3 Mark]
Answer: Let A(9, 2), B(-3, -2)
given the line joining the points AB is divided by y - axis, Let it be in the ratio m:n.
Let that point Xn be P(x, y) then
P(x, y) lies on y - axis, hence x = 0, P(0, y)
0 = -3m + 9n
3m = 9n
m = 3n
m:n = 3:1
add 1 to both the sides,
Point of intersection = (0, -1)
Question 20
Question: Show that the co-ordinates of the point dividing the join of the points (3, -5) and (-4, 6) in the ratio of 2:3 and the join of points (-4, 6) and (3, -5) in the ratio of 3:2 are the same. [3 Mark]
Answer: Let P(x, y) be the point dividing the join points A(3, -5) and (-4, 6) in the ratio 2:3.
According to section formula,
Let Q (a, b) be the point which divides the line joining the points in the ratio 3:2.
Let m = 3, n= 2, points are (-4, 6), (3, -5).
By section formula,
Hence the coordinates of point P(x, y) = Point Q
