Question-1 :- Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3.
Solution :-Let P(x, y) be the required point. Given that : x₁ = -1, y₁ = 7 ; x₂ = 4, y₂ = -3 ; m₁ = 2, m₂ = 3 Using the section formula, we getx = [(2x4 + 3x(-1))/(2 + 3)] = (8 - 3)/5 = 5/5 = 1 y = [(2x(-3) + 3x7)/(2 + 3)] = (-6 + 21)/5 = 15/5 = 3 Therefore, (1, 3) is the required point.
Question-2 :- Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).
Solution :-Let P and Q be the points of trisection of AB i.e., AP = PQ = QBTherefore, P divides AB internally in the ratio 1 : 2. Therefore, the coordinates of P, by applying the section formula, are [(1x(-2) + 2x4)/(1 + 2), (1x(-3) + 2x(-1))/(1 + 2)] i.e., [(-2+8)/3, (-3-2)/3] i.e., (3, -5/3) Now, Q also divides AB internally in the ratio 2 : 1. So, the coordinates of Q are [(2x(-2) + 1x4)/(2 + 1), (2x(-3) + 1x(-1))/(2 + 1)] i.e., [(-4+4)/3, (-6-1)/3] i.e., (0, -7/3) Therefore, the coordinates of the points of trisection of the line segment joining A and B are (–1, 0) and (– 4, 2).
Question-3 :- To conduct Sports Day activities, in your rectangular shaped school ground ABCD,
lines have been drawn with chalk powder at a distance of 1m each.
100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. 7.12.
Niharika runs 1/4th the distance AD on the 2nd line and posts a green flag.
Preet runs 1/5th the distance AD on the eighth line and posts a red flag.
What is the distance between both the flags?
If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?
It can be observed that Niharika posted the green flag at 1/4 of the distance AD i.e., (1/4 x 100)m = 25m from the starting point of 2nd line. Therefore, the coordinates of this point G is (2, 25). Similarly, Preet posted red flag at 1/5 of the distance AD i.e., (1/5 x 100)m = 20m from the starting point of 8th line. Therefore, the coordinates of this point R are (8, 20). Distance between these flags by using distance formula = GR = √(8-2)² + (25-20)² = √(6)² + (5)² = √61 m The point at which Rashmi should post her blue flag is the mid-point of the line joining these points. Let this point be A (x, y). x = (2+8)/2, y = (25+20)/2 x = 5, y = 22.5 Hence, A(x, y) = (5, 22.5) Therefore, Rashmi should post her blue flag at 22.5m on 5th line
Question-4 :- Find the ratio in which the line segment joining the points (–3, 10) and (6, – 8) is divided by (– 1, 6).
Solution :-
Let the ratio in which the line segment joining (−3, 10) and (6, −8) is divided by point (−1, 6) be k : 1.
Therefore,
(-1, 6) = [(6k - 3)/(k + 1), (-8k + 10)/(k + 1)]
-1 = (6k - 3)/(k + 1)
-k - 1 = 6k - 3
-k - 6k = -3 + 1
-7k = -2
k = 2/7
Therefore, Required ratio is 2:7.
Question-5 :- Find the ratio in which the line segment joining A(1, – 5) and B(–4, 5) is divided by the x-axis. Also find the coordinates of the point of division.
Solution :-
Let the ratio be k : 1.
Then by the section formula, the coordinates of the point which divides AB in the ratio k : 1 are
[(-4k + 1)/(k + 1), (5k - 5)/(k + 1)]
This point lies on the x-axis, and we know that on the x-axis the ordinate is 0.
Therefore,
(5k - 5)/(k + 1) = 0
5k - 5 = 0
5k = 5
k = 1
i.e., the ratio is 1 : 1.
Putting the value of k = 1, we get the point of intersection as
= [(-4 + 1)/2, (5 - 5)/2]
= (-3/2, 0)
Question-6 :- If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
Solution :-Let (1, 2), (4, y), (x, 6), and (3, 5) are the coordinates of A, B, C, D vertices of a parallelogram ABCD. Intersection point O of diagonal AC and BD also divides these diagonals.Therefore, O is the mid-point of AC and BD. If O is the mid-point of AC, then the coordinates of O are [(1+x)/2, (2+6)/2] = [(x+1)/2, 4] If O is the mid-point of BD, then the coordinates of O are [(4+3)/2, (5+y)/2] = [7/2, (y+5)/2] Since both the coordinates are of the same point O, (x+1)/2 = 7/2 ; 4 = (y+5)/2 x + 1 = 7 ; 8 = y + 5 x = 7 - 1 ; y = 8 - 5 x = 6; y = 3
Question-7 :- Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, –3) and B is (1, 4).
Solution :-
Let the coordinates of point A be (x, y).
Mid-point of AB is (2, −3), which is the center of the circle.
Therefore,
(2, -3) = [(x+1)/2, (y+4)/2]
2 = (x+1)/2 ; -3 = (y+4)/2
4 = x + 1 ; -6 = y + 4
x = 4 - 1 ; y = -6 - 4
x = 3; y = -10
Hence, the coordinates of A are (3, -10).
Question-8 :- If A and B are (–2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = 3AB/7 and P lies on the line segment AB.
Solution :-
The coordinates of point A and B are (−2, −2) and (2, −4) respectively.
AP = 3AB/7
AP/AB = 3/7
PB = AB - AP = 7 - 3 = 4
So, AP/PB = 3/4
Therefore, AP: PB = 3:4
Point P divides the line segment AB in the ratio 3:4.
Coordinates of P
= [(3x2 + 4x(-2))/(3 + 4), (3x(-4) + 4x(-2))/(3 + 4)]
= [(6 - 8)/7, (-12 - 8)/7]
= (-2/7, -20/7)
Question-9 :- Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.
Solution :-From the figure, it can be observed that points P, Q, R are dividing the line segment in a ratio 1:3, 1:1, 3:1 respectively.Now, Coordinates of P = [(1x2 + 3x(-2))/(1 + 3), (1x8 + 3x2)/(1 + 3)] = (-1, 7/2) Also, Coordinates of Q = [(1x2 + 1x(-2))/(1 + 1), (1x8 + 1x2)/(1 + 1)] = (0, 5) Also, Coordinates of R = [(3x2 + 1x(-2))/(3 + 1), (3x8 + 1x2)/(3 + 1)] = (1, 13/2)
Question-10 :- Find the area of a rhombus if its vertices are (3, 0), (4, 5), (– 1, 4) and (– 2, – 1) taken in order.
Solution :-Let (3, 0), (4, 5), (−1, 4) and (−2, −1) are the vertices A, B, C, D of a rhombus ABCD.Now, Length of diagonal of AC = √(3+1)² + (0-4)² = √(4)² + (4)² = √32 = 4√2 Also, Length of diagonal of BD = √(4+2)² + (5+1)² = √(6)² + (6)² = √72 = 6√2 Therefore, Area of Rhombus ABCD = 1/2 x 4√2 x 6√2 = 24 square units.
