Class 10 maths NCERT solutions of chapter 7- Coordinate Geometry
Class 10 maths NCERT solutions of chapter 7- Coordinate Geometry are presented here for helping the students of class 10 in their preparation of maths paper in SA-1 and SA-2 CBSE board exams. These NCERT solutions of chapter 7-Coordinate geometry will help you a complete understanding of the concept of coordinate geometry required to solve all the questions of co-ordinate geometry involved in the 10 class maths question paper of the CBSE board 2020-21. All questions of chapter 7- coordinate geometry are solved by an expert of maths as per the CBSE norms and by the step by step method so that every student could understand the solutions.
Class 10 maths NCERT solutions of chapter 7- Coordinate Geometry
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EXERCISE 7.1
Q1. Find the distance between the following pairs of points:
(i) (2, 3), (4, 1)
(ii) (-5, 7), (-1, 3)
(iii) (a, b), (-a, -b
Ans. (i) Let the points are denoted by A(2, 3) and B(4, 1)
Applying the distance (d) formula between two points (x_{1,} y_{1}) and (x_{2}, y_{2})
Here x_{1} =2, y_{1}= 3 and x_{2}=4 ,y_{2} = 1
(ii) A(-5, 7), B(-1, 3)
x_{1} = -5, y_{1}= 7 and x_{2}= -1 ,y_{2} = 3
x_{1} = a, y_{1}= b and x_{2}= -a ,y_{2} = -b
Q2.Find the distance between the points (o,0) and (36,15). Can you now find the distance between the two towns A and B discussed in section 7.2.
Ans. Let the town A (0,0) and town B(36,15), so the distance (d) between two points is
x_{1} = 0, y_{1}= 0 and x_{2}= 36 ,y_{2} = 15
So, the distance between two towns A and B is = 39 units
Q3.Determine if the points (1,5), (2,3) and (-2,-11) are collinear.
Ans. Let the points are denoted by A (1,5), B(2,3) and C(-2,-11)
If the points A, B and C are collinear then they should have to satisfy the following
AB + BC = AC
The distance between A (1,5), B(2,3)
x_{1} = 1, y_{1}= 5 and x_{2}= 2 ,y_{2} = 3
AB + BC ≠AC
Hence A, B, and C are not collinear
Q4.Check whether (5,-2), (6,4), and (7, -2) are the vertices of an isosceles triangle.
Let the vertices of the triangle are A(5,-2), B(6,4), and C(7, -2)
AB = BC
Therefore (5,-2), (6,4), and (7, -2) are the vertices of an isosceles triangle
Q5. In a classroom, 4 friends are seated at points A, B, C, and D as shown in the given figure. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct.
You can see the video -Class 10 Maths NCERT solution of Question no-5 Exercise 7.1 Chapter 7 Coordinate Geometry
Ans. It is seen in the figure that the coordinates of the A, B, C and are follows
A(3,4), B(6,7), C(9,4) and D(6,1)
For checking ABCD is a square, let’s check the length of 4 sides and diagonals by applying the distance formula
So, AB = BC = CD = AD
Now the calculating length of the diagonals AC and BD
The coordinates of both ends of the diagonal AC are (3,4) and (9,4)
The coordinates of both ends of the diagonal BD are (6,7) and (7,1)
The length of both diagonals are also equal
Since all sides and diagonals of ABCD are equal, so Champa is correct that ABCD is a square.
Q6.Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer.
(i) (-1, -2), (1, 0), (-1, 2), (-3, 0)
(ii) (-3, 5), (3, 1), (0, 3), (-1, -4)
(iii) (4, 5), (7, 6), (4, 3), (1, 2)
Ans. (i) Let the vertices of the quadrilateral are A (-1, -2), B(1, 0), C(-1, 2) and D(-3, 0)
Determining the length of the sides AB, BC, CD, AD, and diagonals AC and BD by applying the distance formula
The length of all sides AB = BC = CD = AD = 2√2 and length of diagonals AC = BD = 4,therefore ABCD is a square.
(i) Let the vertices of the quadrilateral are A (-3, 5), B(3, 1), C(0, 3) and D(-1, -4)
Determining the length of the sides AB, BC, CD, AD by applying the distance formula
All sides are not equal, therefore the given points are vertices of a simple quadrilateral
(iii) Let the vertices of the quadrilateral are A(4, 5), B(7, 6), C(4, 3), D(1, 2)
Determining the length of the sides AB, BC, CD, AD by applying the distance formula
AB = CD = √10 and BC = AD = 3√2
Opposites sides of the quadrilateral are equal therefore ABCD is a parallelogram
Q7. Find the point on the x-axis which is equidistant from (2, -5) and (-2, 9).
You can see the Video also
Ans. Let the point on the x-axis is A(x, 0) which is equidistant from B(2, -5) and C(-2, 9)
∴ AB = AC
The distance between A(x,0) and B(2, -5) = The distance between A(x,0) and C(-2, 9)
x² – 4x +29 = x ² + 4x + 85
8x = – 5 6
x = -7
Hence the point on the x-axis is A(-7, 0) which is equidistant from B(2, -5) and C(-2, 9)
Q8. Find the values of y for which the distance between the points P (2, -3) and Q (10, y) is 10 units.
Ans. The distance between the points P (2, -3) and Q (10, y) = 10 units
64 + 9 + y² + 6y = 100
y² + 6y – 27 = 0
y² + 9y – 3y – 27 = 0
y(y + 9) – 3( y + 9) = 0
(y + 9)( y – 3) = 0
y = -9, y = 3
Hence the required value of y is – 9 or 3
Q9. If Q (0, 1) is equidistant from P (5, -3), and R (x, 6), find the values of x. Also, find the distances QR and PR.
Ans. We are given that Q (0, 1) is equidistant from P (5, -3), and R (x, 6)
The distance between Q(0,1) and P(5, -3) = The distance between Q(0,1) and R(x, 6)
25 +16 = x² + 25
x = ± 4
The coordinates of Q are (0,1) and of R are ( ± 4, 6)
The coordinates of P are (5,-3) and of R are ( ± 4, 6)
Taking P(5, -3) and R( 4, 6)
Taking P(5, -3) and R( -4, 6)
Therefore QR = √41 and PR = √82, 9√2
Q10.Find a relation between x and y such that the point (x, y) is equidistant from the points (3, 6) and (-3, 4).
Ans. It is given to us that the point (x, y) is equidistant from the points (3, 6) and (-3, 4)
The distance between (x.y) and (3,6) = The distance between (x,y) and (-3,4)
Squaring both sides
(x – 3)² + ( y – 6)² = (x + 3)² + ( y – 4)²
x² + 9 – 6x + y² + 36 – 12y = x² + 9 + 6x + y² + 16 – 8y
– 6x – 6x – 12y + 8y + 45 – 25 = 0
– 12x – 4y + 20 = 0
–4(3x + y – 5) = 0
3x + y – 5 = 0
Hence required relationship between x and y is 3x + y – 5 = 0
Class 10 maths NCERT solutions of chapter 7- Coordinate Geometry
EXERCISE 7.2
Q1. Find the coordinates of the point which divides the join of (- 1, 7) and (4, – 3) in the ratio 2:3.
Ans. Let the coordinates of the point C are (x,y) which divides the join of (- 1, 7) and (4, – 3)
You can see the video -Class 10 Maths NCERT solution of Question no-1 Exercise 7.2 Chapter 7 Coordinate Geometry
Applying the section formula
x_{1} = –1, y_{1}= 7 , x_{2 }= 4, y_{2}= – 3, m_{1} = 2, m_{2} = 3
x = 1, y =3
Hence the coordinates of C are (1,3)
Q2.Find the coordinates of the points of trisection of the line segment joining (4, -1) and (-2, -3).
Solution:
Ans. Let P(a,b) and Q(c,d) trisect the line segment joining A(4, -1) and B(-2, -3)
Applying the section formula for coordinates of P(a,b) that divides the given line segment in 1 : 2
x_{1} = 4, y_{1}= -1, x_{2 }= -2, y_{2}= – 3, m_{1} = 1, m_{2} = 2 and (x,y) = (a, b)
a = 2, b = –5/3, Coordinates of P are ( 2, -5/3)
Co-ordinates of Q are 1/2(2 -2),1/2[(-5/3)-3] =(0,-10/3)
Q3.(c,d) is the mid points of the line segment joing P(a,b) and B(-2,-3), (a,b) = (2, -5/3)
c = 0, d = -7/3
So, coordinates of Q are (0, -7/3)
Hence the points P(2, -5/3) and Q(0. -7/3) trisects the line segment AB.
Q3.To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along with AD, as shown in the following figure. Niharika runs 1/4 th 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?
Ans. We are given distance AD = 100 m and 100 flower pots which are placed at a distance of 1 m.
Niharika placed the flag on the second line at a distance of
Niharika placed the flag at a distance of 25 m from the starting point means on the line of 25 th flower pot, so the coordinates of the point Niharika placed the flag are (2, 25)
Preet placed the flag on the eighth line at a distance of
Niharika placed the flag at a distance of 20 m from the starting point means on the line of 20 th flower pot, so the coordinates of the point Niharika placed the flag are (8, 20)
The distance between both of the flags located at(2,25) and (8,20) is
So, the distance between both of the flags is √61 m.
Rashmi placed the flag at the midpoint of both of these flags located at (2, 25) and (8, 20), let the coordinates of the point at which Rashmi posted the flag are (x, y).
Rashmi placed her blue flag on 5 th line at a distance of 22.5 m.
Q4.Find the ratio in which the line segment joining the points (-3, 10) and (6, – 8) is divided by (-1, 6).
Ans. Applying the section formula
x_{1} = -3, y_{1}= 10 and x_{2}= 6 ,y_{2} = -8, x = -1, y = 6 and k= m_{1}/m_{2} where (m_{1} : m_{2}) is the ratio at which (-1, 6) point divides the line segment
-k -1 = 6k – 3, 6k + 6 = -8k + 10
-7k = -2, 14 k = 4
k = 2/7, k = 2/7
Both of the equation confirms that the point (-1, 6) divides the line segment in 2 : 7.
Q5. 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.
Ans. Let the point on x-axis at which the line segment joining A (1, – 5) and B (- 4, 5) is divided by the x-axis is = (x, 0)
We are having the known value of y coordinate of (x,0) = 0, so applying the section formula accordingly.
y_{1}= -5 , y_{2} = 5, y = 0
5k – 5 = 0
k = 1
Hence the ratio at which the line segment is divided by x axis is 1 : 1.
Let the coordinates of the point of division are (x,y)
x_{1} = 1, y_{1}= -5 and x_{2}= -4 ,y_{2} = 5, (x , y) = (x,0), k = 1
Hence the coordinates of point of division are (-3/2, 0)
Q6. If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
Ans. Let the vertices of a parallelogram are A(1, 2), B(4, y), C(x, 6) and D(3, 5)
Since the diagonals of a parallelogram bisect each other
The midpoint of AC = Midpoint of BD
x = 6
y = 3
Hence the value of x = 6 and of y = 3
Q7.Find the coordinates of point A, where AB is the diameter of a circle whose centre is (2, – 3) and B is (1,4).
Ans. AB is the diameter of the circle and the coordinates of B and of centre are given (1,4) and (2, -3) respectively.
Let the coordinates of another end(A) of diameter is = (x, y)
(2, -3) is the centre of circle so it is the midpoint of AB
x = 3
y = -6 – 4 = -10
Hence the coordinates of A are (3, -10)
Q8. If A and B are (-2, -2) and (2, -4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.
Ans.
We are given that
Subtracting 1 from both sides
AB – AP = BP
AP: BP = 3 : 4
Coordinates of both ends of line segments are A(-2,-2) and B(2, -4) . Let coordinates of P are (x, y) that divides AB in 3 : 4.
Applying the section formula
x_{1} = -2, y_{1}= -2 and x_{2}= 2 ,y_{2} = -4, ,m_{1} = 3, m_{2} = 4
Hence the coordinates of P are (-2/7, -20/7)
Q9.Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.
Ans.Drawing a line segment AB in which P, Q and R points divide it into four equal parts
It is seen in the figure that P , Q, and R divides AB in the ratio of 1 : 3, 1: 1 and 3: 1 respectively
For calculating the coordinates of P (x,y) , applying section formula
x_{1} = -2, y_{1}= 2 and x_{2}= 2 ,y_{2} = 8, ,m_{1} = 1, m_{2} = 3
x = -1, y = 7/2
So, coordinates of P are (-1, 7/2)
Q is the midpoint of AB, the coordinates of A and B are (-2,2) and (2,8)
So, coordinates of Q = 1/2(-2 +2), 1/2(2 + 8)= (0,5)
R is the mid point of BQ, the coordinates of B and Q are (2. 8) and (0,5)
So, coordinates of R = 1/2(2 +0), 1/2(8 + 5) = (1, 13/2)
Hence the coordinates of P, Q and R are (-1, 7/2), (0,5) and (1, 13/2) respectively
Q10.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 = 1/2(product of its diagonals)]
Ans. Let the vertices of the rhombus are A(3,0), B(4,5), C(-1,4) and D(-2, -1) respectively
Area of rhombus = 1/2(Product of diagonals)
So, let’s found out the length of the diagonals AC and BD by applying the distance formula
Area of rhombus ABCD = 1/2(AC× BD) = 1/2( 4√2 × 6√2) = 1/2(24 × 2) = 24 Sq.unit
Class 10 maths NCERT solutions of chapter 7- Coordinate Geometry
EXERCISE 7.3
Q1. Find the area of the triangle whose vertices are:
(i) (2, 3), (-1, 0), (2, -4)
(ii) (-5, -1), (3, -5), (5, 2)
Ans. Area of Δ
(i) Let vertices of the triangles are A(2, 3), B(-1, 0) and C(2, -4)
x_{1} = 2, y_{1} = 3, x_{2}= -1, y_{2} = 0, x_{3} = 2, y_{3} = – 4
Substitute all the values in the above formula, we get
Area of Δ
Therefore area of the triangle is = 10.5 sq.unit
(ii) Let vertices of the triangles are A(-5, -1) , B (3, -5) and C(5, 2)
x_{1} = -5, y_{1} = -1, x_{2}= 3, y_{2} = -5, x_{3} = 5, y_{3} = 2
Therefore area of triangle is = 32 sq.unit
Q2. In each of the following find the value of ‘k’, for which the points are collinear.
(i) (7, -2), (5, 1), (3, -k)
(ii) (8, 1), (k, -4), (2, -5)
Ans. The points will be collinear if the area formed by these points is zero, since the figure formed by the three points must be a triangle, so applying the formula of triangle.
(i) x_{1} = 7, y_{1} = -2, x_{2}= 5, y_{2} = 1, x_{3} = 3, y_{3} = -k
7(1 +k) + 5(-k + 2) + 3( -2 -1) = 0
7k -5k + 7 + 10 – 9 = 0
2k + 8 = 0
k = – 4
(ii) The points will be collinear if the area of the triangle formed by the points (8, 1), (k, -4), (2, -5) is zero.
x_{1} = 8, y_{1} = 1, x_{2}= k, y_{2} = -4, x_{3} = 2, y_{3} = -5
8 – 6k +10 = 0
-6k = -18
k = 3
Q3.Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.
Ans. Let the verteces of the triangle are A(0, -1), B (2, 1) and C(0, 3)
The cordinates of A and B are (0, -1) and (2, 1)
So, the mid point of AB = 1/2(0 + 2)/2, 1/2(-1 + 1)/2 = D(1, 0)
The cordinates of B and C are (2, 1) and (0, 3)
So,the mid point of BC are = 1/2(2+0), 1/2(1 +3) = F(1 , 2)
The cordinates of A and C are (0, -1) and (0, 3)
So,the mid point of AC are = 1/2(0 +0), 1/2(-1 +3) = E(0 , 1)
The vertices of the ΔABC are (0, -1), (2, 1) and (0, 3)
x_{1} = 0, y_{1} = -1, x_{2}= 2, y_{2} = 1, x_{3} = 0, y_{3} = 3
arΔABC = 4 sq.unit
The vertices of the ΔDEF are (1, 0), (0, 1) and (1, 2)
x_{1} = 1, y_{1} = 0, x_{2}= 0, y_{2} = 1, x_{3} = 1, y_{3} = 2
arΔDEF = -1 , avoiding minus sign because area can’t be negative
arΔDEF = 1 square unit
The ratio of ΔDEF to ΔABC
Therefore the required ratio of both triagles is 1 : 4
Q4.Find the area of the quadrilateral whose vertices, taken in order, are
(-4, -2), (-3, -5), (3, -2) and (2, 3).
Ans. Let the vertices of quadrilatral are A(-4, -2), B(-3, -5), C(3, -2) and D(2, 3)
After we draw a diagonal AC, we get two triangles ΔABC and ΔADC
The triangle ABC has the vertices ( -4, -2), (-3, -5) and (3, -2 )
x_{1} = -4, y_{1} = -2, x_{2}= -3, y_{2} = -5, x_{3} = 3, y_{3} = -2
So, area of ABC is = 10.5 sq.unit
The triangle ADC has the vertices ( -4, -2), (2, 3) and (3, -2 )
x_{1} = -4, y_{1} = -2, x_{2}= 2, y_{2} = 3, x_{3} = 3, y_{3} = -2
Avoiding minus sign because area can’t be negative
arΔADC = 35/2 = 17.5 sq,unit
Area of given quadrilateral ABCD = ar ΔABC + ar ΔADC = 10.5 + 17.5 = 28 sq.unit
Q5.You have studied in Class IX that a median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC whose vertices are A (4, – 6), B (3, – 2) and C (5, 2).
Ans. We are given the vertices of the triangle A (4, – 6), B (3, – 2) and C (5, 2)
Let CD is the mediun of the triagle ABC, so D is the mid point of AB
The coordinates of D are = 1/2(4 +3), 1/2(-6 -2) = (7/2, -4)
We have to show the area of triagle ADC = area of triagle DBC
The verteces of the triagle ADC are (4, -6), (5, 2), C(7/2, -4)
x_{1} = 4, y_{1} = -6, x_{2}= 5, y_{2} = 2, x_{3} = 7/2, y_{3} = -4
Area of triagle
The vertices of the triangle DBC are (7/2, -4), (3, -2), C(5, 2)
x_{1} = 7/2, y_{1} = -4, x_{2}= 3, y_{2} = -2, x_{3} = 5, y_{3} = 2
Avoiding minus sign because area can’t be negative
So, area of triangle DBC = 3 sq.unit
Hence area of ADC = area of DBC
Therefore mediun CD divides the triangle ABC into two equal parts.
Class 10 maths NCERT solutions of chapter 7- Coordinate Geometry
Exercise 7.4
Q1. Determine the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A(2, –2) and B(3, 7).
Ans. Let the given line divides the line segment in the ratio of (m_{1} : m_{2}) and the point of intersection is x, y
x_{1} = 2, x_{2}= 3, y_{1} = -2, y_{2} = 7
Since the line 2x + y – 4 = 0 passes through the poins(x, y), so the coordinates will satisfy the equation.
_{}
Therefore the given line divides the line segment in 2 : 9
Q2. Find the relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.
Ans. If the given points are collinear then the area of the triangle formed by these points must be zero
x_{1} = x, x_{2}= 1, y_{1} = y, y_{2} = 2, x_{3} = 7, y_{3} = 0
2x – y + 7y -14 = 0
2x + 6y = 14
x + 3y = 7
Therefore the required relationship between x and y is x + 3y = 7
Q3. Find the centre of a circle passing through points (6, -6), (3, -7) and (3, 3).
Ans. Let the centre of circle is (x, y), the distance of the centre to the circle is known as radii
So, the distance between (x,y) and (6, -6) = The distance between (x,y) and (3, -7)
x² + 6² – 12x + y² + 6² + 12x = x² + 3² – 6x + y² + 7² + 14y
72 = -6x + 14y + 9 + 49
6x – 14y + 14 = 0
3x – 7y + 7 = 0…….(i)
The distance between (x,y) and (6, -6) = The distance between (x,y) and (3, 3)
x² + 6² – 12x + y² + 6² + 12x = x² + 3² – 6x + y² + 3² – 6y
72 = -6x – 6y + 18
6x + 6y + 54 = 0
x + y + 9 = 0…….(ii)
Multiplying eq. by 3 , we get eq.(iii)
3x + 3y + 27 = 0……(iii)
Subtracting eq.(i) from (iii)
10y + 20 = 0
y = -2
Putting the value of y = -2 in eq.(1)
x = 3
Therefore coordinates of the centre of the circle is (3, -2)
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