Class 10 - Mathematics
Coordinate Geometry - Exercise 7.2
Top Block 1
Exercise 7.2
Question : 1:Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3.
Answer :
Let P(x, y) be the required point. Using the section formula, we obtain
x = {2 * 4 + 3 * (-1)}/(2 + 3) = (8 – 3)/5 = 5/5 = 1
y = {2 * (-3) + 3 * 7}/(2 + 3) = (-6 + 21)/5 = 15/5 = 3
Therefore, the point is (1, 3).
Question : 2:Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).
Answer :
Let P (x1, y1) and Q (x2, y2) are the points of trisection of the line segment joining the given
points i.e., AP = PQ = QB
Therefore, point P divides AB internally in the ratio 1 : 2.
Now, x1 = {1 * (-2) + 2 * 4}/(1 + 2) = (-2 + 8)/5 = 6/3 = 2
y1 = {1 * (-3) + 2 * (-1)}/(1 + 2) = (-3 – 2)/3 = -5/3
Therefore, P (x1, y1) = (2, 5/3)
Point Q divides AB internally in the ratio 2 : 1.
Now, x2 = {2 * (-2) + 1 * 4}/(2 + 1) = (-4 + 4)/5 = 0
y2 = {2 * (-3) + 1 * (-1)}/(2 + 1) = (-6 – 1)/3 = -7/3
Therefore, Q (x2, y2) = (0, -7/3)
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?
Mddle block 1
Answer :
It can be observed that Niharika posted the green flag at of the distance AD i.e., (1/4 * 100)
= 25 m from the starting point of 2nd line. Therefore, the coordinates of this point G is (2, 25).
Similarly, Preet posted red flag at of the distance AD i.e., m from the (1/5 * 100) = 25 m 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 – 20)2 + (25 – 20)2} = √ (36 + 25) = √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).
Now, x = (2 + 8)/2 = 12/2 = 5
and y = (25 + 20)/2 = 45/2 = 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).
Answer :
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 = (6k – 3)/(k + 1)
⇒ -(k + 1) = 6k – 3
⇒ -k – 1 = 6k – 3
⇒ 3 – 1 = 6k + k
⇒ 7k = 2
⇒ k = 2/7
Hence, the required ratio = 2/7 : 1 = 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.
Answer :
Let the ratio in which the line segment joining A (1, −5) and B (−4, 5) is divided by x-axis be
K : 1.
Therefore, the coordinates of the point of division = {(-4k + 1)/(k + 1), (5k – 5)/(k + 1)}
We know that y-coordinate of any point on x-axis is 0.
⇒ (5k – 5)/(k + 1) = 0
⇒ 5k – 5 = 0
⇒ 5k = 5
⇒ k = 5/5
⇒ k = 1
Therefore, x-axis divides it in the ratio 1 : 1.
Division point = {(-4 * 1 + 1)/(1 + 1), (5 * 1 – 5)/(1 + 1)}
= {(-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.
Answer :
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} = {(1 + x)/2, 4}
If O is the mid-point of BD, then the coordinates of O are
{(4 + 3)/2, (5 + y)/2} = {7/2, (5 + y)/2}
Since both the coordinates are of the same point O,
So, (1 + x)/2 = 7/2 and 4 = (5 + y)/2
⇒ 1 + x = 7 and 5 + y = 8
⇒ x = 6 and 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).
Answer :
Let the coordinates of point A be (x, y).
Mid-point of AB is (2, −3), which is the center of the circle.
So, (2, -3) = {(x + 1)/2, (y + 4)/2]
⇒ (x + 1)/2 = 2 and (y + 4)/2 = -3
⇒ x + 1 = 4 and y + 4 = -6
⇒ x = 3 and y = -10
Therefore, the coordinate of A is (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.
Answer :
Since AP = 3AB/7
⇒ AP/AB = 3/7
Therefore, AP : PB = 3 : 4
Point P divides the line segment AB in the ratio 3 : 4.
Now, coordinate of P = [{3 * 2 + 4 * (-2)}/(3 + 4), {3 * (-4) + 4 * (-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.
Answer :
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.
= {(2 – 6)/4, (8 + 6)/4}
= (-4/4, 14/4)
= (-1, 7/2)
Coordinate of Q = [{2 + (-2)}/2, (2 + 8)/2] = (0, 10/2) = (0, 5)
Coordinate of R = [{3 * 2 + 1 * (-2)}/(3 + 1), {3 * 8 + 1 * 2}/(3 + 1)]
= {(3 – 2)/4, (24 + 2)/4}
= (1/4, 26/4)
= (1/4, 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. [Hint: Area of a rhombus = (product of its diagonals)/2]
Answer :
Length of diagonal AC = √[{3 – (-1)}2 + (0 – 4)2]
= √[(3 + 1)2 + (- 4)2]
= √(16 + 16)
= √32
= 4√2
Length of diagonal BD = √[{4 – (-2)}2 + {5 – (-1)}2]
= √[(4 + 2)2 + (5 + 1)2]
= √(36 + 36)
= √72
= 6√2
Therefore, area of rhombus = (1/2) * AC * BD
= (1/2) * 4√2 * 6√2
= 48/2
= 24 square units
