Chapter 7 - Co-ordinate Geometry
Practice Set 7.1
1. State in which quadrant or on which axis do the following points lie.
(i) A(-3, 2)
Solution:
The x coordinate of A(-3, 2) is negative, and its y coordinate is positive.
Therefore, point A (-3, 2) is in the second quadrant.
(ii) B(-5, -2)
Solution:
The x coordinate of B(-5, -2) is negative, and its y coordinate is also negative.
Therefore, point B (-5, -2) is in the third quadrant.
(iii) K(3.5, 1.5)
Solution:
The x coordinate of K (3.5, 1.5) is positive, and its y coordinate is positive.
Therefore, point K (3.5, 1.5) is in the first quadrant.
(iv) D(2, 10)
Solution:
The x coordinate of D(2, 10) is positive, and its y coordinate is also positive.
Therefore, point D (2, 10) is in the first quadrant.
(v) E(37, 35)
Solution:
The x coordinate of E (37, 35) is positive, and its y coordinate is also positive.
Therefore, point E (37, 35) is in the first quadrant.
(vi) F(15, -18)
Solution:
The x coordinate of F (15, -18) is positive, and its y coordinate is negative.
Therefore, point F (15, 18) is in the fourth quadrant.
(vii) G(3, -7)
Solution:
The x coordinate of G(3, -7) is positive, and its y coordinate is negative.
Therefore, point G (3, -7) is in the fourth quadrant.
(viii) H(0, -5)
Solution:
The x coordinate of H (0, -5) is zero.
Therefore, point H (0, -5) is on the Y-axis.
(ix) M(12, 0)
Solution:
The y coordinate of M (12, 0) is zero.
Therefore, point M (12, 0) is on the X-axis.
(x) N(0, 9)
Solution:
The x coordinate of N (0, 9) is zero.
Therefore, point N (0, 9) is on the Y-axis.
(xi) P(0, 2.5)
Solution:
The x coordinate of P (0, 2.5) is zero.
Therefore, point P(0, 2.5) is on the Y-axis.
(xii) Q(-7, -3)
Solution:
The x coordinate of Q (-7, -3) is negative, and its y coordinate is also negative.
Therefore, point Q (-7, -3) is in the third quadrant.
2. In which quadrant are the following points ?
(i) whose both co-ordinates are positive.
Solution:
The x and y coordinates of a point are both positive in the first quadrant.
Therefore, the points are in the first quadrant.
(ii) whose both co-ordinates are negative.
Solution:
The x and y coordinates of a point are both negative in the third quadrant.
Therefore, the points are in the third quadrant.
(iii) whose x co-ordinate is positive, and the y co-ordinate is negative.
Solution:
The x coordinate of a point is positive and the y coordinate of a point is negative in the fourth quadrant.
Therefore, the points are in the fourth quadrant.
(iv) whose x co-ordinate is negative and y co-ordinate is positive.
Solution:
The x coordinate of a point is negative and the y coordinate of a point is positive in the second quadrant.
Therefore, the points are in the second quadrant.
3. Draw the co-ordinate system on a plane and plot the following points.
L(-2, 4)
M(5, 6)
N(-3, -4)
P(2, -3)
Q(6, -5)
S(7, 0)
T(0, -5)
Solution:
Practice Set 7.2
1. On a graph paper plot the points A (3, 0), B(3, 3), C(0, 3). Join A, B and B, C. What is the figure formed ?
Solution:
The graph of the given points are;
Ans: The figure formed is a square.
2. Write the equation of the line parallel to the Y-axis at a distance of 7 units from it to its left.
Solution:
We know that the equation of a line parallel to the Y-axis is x = a.
And the equation of a line parallel to and left of Y-axis will be x = –a.
According to the given condition, the line is at a distance of 7 units to the left of the Y-axis.
∴ The equation will be x = –7
Ans: x = –7 is the equation of the required line.
3. Write the equation of the line parallel to the X-axis at a distance of 5 units from it and below the X-axis.
Solution:
The equation of a line parallel to the X-axis is y = b.
And the equation of a line parallel to and below the X-axis will be y = –b.
Since, the line is at a distance of 5 units below the X-axis.
∴ The equation will be b = –5
Ans: y = –5 is the equation of the required line.
4. The point Q( -3, -2) lies on a line parallel to the Y-axis. Write the equation of the line and draw its graph.
Solution:
We know that the equation of a line parallel to the Y-axis is x = a.
Here, x = –3
∴ The equation will be x = –3
And the graph of the line will be:
5. Y-axis and line x = –4 are parallel lines. What is the distance between them?
Solution:
Equation of the Y-axis is x = 0.
Equation of the line parallel to the Y-axis is x = – 4 …[Given]
∴ Distance between the Y-axis and the line (x = – 4) = 0 – (– 4) …[Distance cannot be negative, 0 > -4]
∴ Distance between the Y-axis and the line (x = – 4) = 0 + 4
∴ Distance between the Y-axis and the line (x = – 4) = 4 units
Ans: The distance between the Y-axis and the line x = – 4 is 4 units.
6. Which of the equations given below have graphs parallel to the X-axis, and which ones have graphs parallel to the Y-axis ?
(i) x = 3
Solution:
The given equation is x = 3
We know that the equation of a line parallel to the Y-axis is x = a.
∴ The line x = 3 is parallel to the Y-axis.
(ii) y – 2 = 0
Solution:
The given equation is y – 2 = 0
∴ y = 2
We know that the equation of a line parallel to the X-axis is y = b.
∴ The line y – 2 = 0 is parallel to the X-axis.
(iii) x + 6 = 0
Solution:
The given equation is x + 6 = 0
∴ x = –6
We know that the equation of a line parallel to the Y-axis is x = a.
∴ The line x + 6 = 0 is parallel to the Y-axis.
(iv) y = –5
Solution:
The given equation is y = –5
We know that the equation of a line parallel to the X-axis is y = b.
∴ The line y = –5 is parallel to the X-axis.
7. On a graph paper, plot the points A(2, 3), B(6, -1) and C(0, 5). If those points are collinear then draw the line which includes them. Write the co-ordinates of the points at which the line intersects the X-axis and the Y-axis.
Solution:
The graph of the given points A(2, 3), B(6, -1) and C(0, 5) is;
From the graph, the line drawn intersects the X-axis at D(5, 0) and the Y-axis at C(0, 5).
8. Draw the graphs of the following equations on the same system of co-ordinates. Write the co-ordinates of their points of intersection.
x + 4 = 0, y – 1 = 0, 2x + 3 = 0, 3y – 15 = 0
Solution:
The given equations are:
(i) x + 4 = 0
∴ x = – 4
(ii) y – 1 = 0
∴ y = 1
(iii) 2x + 3 = 0
∴ 2x = – 3
∴ x = – \( \frac{3}{2} \)
∴ x = – 1.5
(iv) 3y – 15 = 0
∴ 3y = 15
∴ y = \( \frac{15}{3} \)
∴ y = 5
∴ The graph of the four points are;
The co-ordinates of the point of intersection of the lines; x + 4 = 0 and y – 1 = 0 is A (-4, 1).
The co-ordinates of the point of intersection of the lines; y – 1 = 0 and 2x + 3 = 0 is B (-1.5, 1).
The co-ordinates of the point of intersection of the lines; 3y – 15 = 0 and 2x + 3 = 0 is C (-1.5, 5).
The co-ordinates of the point of intersection of the lines; x + 4 = 0 and 3y – 15 = 0 is D (-4, 5).
9. Draw the graphs of the equations given below.
(i) x + y = 2
Solution:
x + y = 2
∴ y = 2 – x
When x = 0,
y = 2 – x
∴ y = 2 – 0
∴ y = 2
∴ (x, y) = (0, 2)
When x = 1,
y = 2 – x
∴ y = 2 – 1
∴ y = 1
∴ (x, y) = (1, 1)
When x = –1,
y = 2 – x
∴ y = 2 – (–1)
∴ y = 2 + 1
∴ y = 3
∴ (x, y) = (–1, 3)
| x | 1 | -1 | |
|---|---|---|---|
y | 2 | 1 | 3 |
(x, y) | (0, 2) | (1, 1) | (-1, 3) |
∴ The graph of the given equation is;
(ii) 3x – y = 0
Solution:
3x – y = 0
∴ y = 3x
When x = 0,
y = 3x
∴ y = 3 × 0
∴ y = 0
∴ (x, y) = (0, 0)
When x = 1,
y = 3x
∴ y = 3 × 1
∴ y = 3
∴ (x, y) = (1, 3)
When x = –1,
y = 3x
∴ y = 3 × –1
∴ y = –3
∴ (x, y) = (–1, –3)
| x | 1 | -1 | |
|---|---|---|---|
y | 3 | -3 | |
(x, y) | (0, 0) | (1, 3) | (-1, -3) |
∴ The graph of the given equation is;
(iii) 2x + y = 1
Solution:
2x + y = 1
∴ y = 1 – 2x
When x = 0
y = 1 – 2x
∴ y = 1 – (2 × 0)
∴ y = 1 – 0
∴ y = 1
∴ (x, y) = (0, 1)
When x = 1
y = 1 – 2x
∴ y = 1 – (2 × 1)
∴ y = 1 – 2
∴ y = –1
∴ (x, y) = (1, –1)
When x = –1
y = 1 – 2x
∴ y = 1 – (2 × –1)
∴ y = 1 – (–2)
∴ y = 1 + 2
∴ y = 3
∴ (x, y) = (–1, 3)
| x | 1 | -1 | |
|---|---|---|---|
y | 1 | -1 | 3 |
(x, y) | (0, 1) | (1, -1) | (-1, 3) |
∴ The graph of the given equation is;
Problem Set 7
1. Choose the correct alternative answer for the following quesitons.
(i) What is the form of co-ordinates of a point on the X-axis ?
(A) (b, b)
(B) (o, b)
(C) (a, o)
(D) (a, a)
OPTION (C) – (a, 0)
(ii) Any point on the line y = x is of the form …..
(A) (a, a)
(B) (o, a)
(C) (a, o)
(D) (a, – a)
OPTION (A) – (a, a)
(iii) What is the equation of the X-axis ?
(A) x = 0
(B) y = 0
(C) x + y = 0
(D) x = y
OPTION (B) – y = 0
(iv) In which quadrant does the point (-4, -3) lie ?
(A) First
(B) Second
(C) Third
(D) Fourth
Solution:
The x coordinate of (-4, -3) is negative, and its y coordinate is also negative.
Therefore, point (5, -3) is in the third quadrant.
OPTION (C) – Third
(v) What is the nature of the line which includes the points (-5, 5), (6, 5), (-3, 5), (0, 5) ?
(A) Passes through the origin
(B) Parallel to Y-axis
(C) Parallel to X-axis
(D) None of these
Solution:
By observing all the points we get to know that the y coordinate of all the points is the same.
∴ The line which passes through the given points is parallel to X-axis.
OPTION (C) – Parallel to X-axis
(vi) Which of the points P (-1, 1), Q (3, -4), R(1, -1), S (-2, -3), T (-4, 4) lie in the fourth quadrant ?
(A) P and T
(B) Q and R
(C) only S
(D) P and R
Solution:
For a point to lie in the fourth quadrant, it’s x coordinate should be positive and y coordinate should be negative.
OPTION (B) – Q and R
2. Some points are shown in the figure 7.11. With the help of it answer the following questions :
(i) Write the co-ordinates of the points Q and R.
Solution:
Q(-2, 2) and R(4, -1)
(ii) Write the co-ordinates of the points T and M.
Solution:
T(0, -1) and M(3, 0)
(iii) Which point lies in the third quadrant ?
Solution:
Point S lies in the third quadrant.
(iv) Which are the points whose x and y co-ordinates are equal ?
Solution:
The x and y co-ordinates of point O are equal.
3. Without plotting the points on a graph, state in which quadrant or on which axis do the following point lie.
(i) (5, -3)
Solution:
The x coordinate of (5, -3) is positive, and its y coordinate is negative. Therefore, point (5, -3) is in the fourth quadrant.
(ii) (-7, -12)
Solution:
The x coordinate of (-7, -12) is negative, and its y coordinate is also negative. Therefore, point (-7, -12) is in the third quadrant.
(iii) (-23, 4)
Solution:
The x coordinate of (-23, 4) is negative, and its y coordinate is positive. Therefore, point (-23, 4) is in the second quadrant.
(iv) (-9, 5)
Solution:
The x coordinate of (-9, 5) is positive, and its y coordinate is positive. Therefore, point (-9, 5) is in the first quadrant.
(v) (0, -3)
Solution:
The x coordinate of (0, -3) is 0.
Therefore, point (0, -3) lies on the Y-axis.
(vi) (-6, 0)
Solution:
The x coordinate of (-6, 0) is 0.
Therefore, point (-6, 0) lies on the X-axis.
4. Plot the following points on the one and the same co-ordinate system.
A (1, 3), B (-3, -1), C (1, -4), D (-2, 3), E (0, -8), F (1, 0)
Solution:
The graph of the following points are:
5. In the graph alongside, line LM is parallel to the Y-axis. (Fig. 7.12)
(i) What is the distance of line LM from the Y-axis ?
Solution:
The distance of line LM from the Y-axis is 3.
(ii) Write the co-ordinates of the points P, Q and R.
Solution:
The coordinates of the points are P(3, 2), Q(3, -1) and R(3, 0).
(iii) What is the difference between the x co-ordinates of the points L and M?
Solution:
The difference between the x co-ordinates of the points L and M is 6.
6. How many lines are there which are parallel to X-axis and have a distance of 5 units?
Solution:
We know that the equation of a line parallel to the X-axis is y = b.
There are 2 lines which are parallel to X-axis and at a distance of 5 units.
The equations of the lines are y = 5 and y = –5.
7*. If ‘a’ is a real number, what is the distance between the Y-axis and the line x = a ?
Solution:
We know that the equation of the Y-axis is x = 0.
Since, ‘a’ is a real number, there can be two possibilities.
Case I: a > 0
Case II: a < 0
∴ Distance between the Y-axis and the line x = a is a – 0 = a
Since,
|a| = a, a > 0
|a| = – a, a < 0
∴ Distance between the Y-axis and the line x = a is |a|.