MAK LABS — Std 9 Maths Part 2 Chapter 7 Co-ordinate Geometry — Practice Sets 7.1, 7.2 & Problem Set 7 Solutions

1State in which quadrant or on which axis do the following points lie. A(−3, 2), B(−5, −2), K(3.5, 1.5), D(2, 10), E(37, 35), F(15, −18), G(3, −7), H(0, −5), M(12, 0), N(0, 9), P(0, 2.5), Q(−7, −3)

Look at the signs of the co-ordinates. If the x co-ordinate is 0 the point is on the Y-axis; if the y co-ordinate is 0 it is on the X-axis.

Point	Signs of (x, y)	Quadrant / axis
A(−3, 2)	−, +	Quadrant II
B(−5, −2)	−, −	Quadrant III
K(3.5, 1.5)	+, +	Quadrant I
D(2, 10)	+, +	Quadrant I
E(37, 35)	+, +	Quadrant I
F(15, −18)	+, −	Quadrant IV
G(3, −7)	+, −	Quadrant IV
H(0, −5)	x = 0	Y-axis
M(12, 0)	y = 0	X-axis
N(0, 9)	x = 0	Y-axis
P(0, 2.5)	x = 0	Y-axis
Q(−7, −3)	−, −	Quadrant III

2In which quadrant are the following points?

Both co-ordinates positive — Quadrant I
Both co-ordinates negative — Quadrant III
x co-ordinate positive, y co-ordinate negative — Quadrant IV
x co-ordinate negative, y co-ordinate positive — Quadrant II

3Draw 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)

For each point, move along the X-axis by the x co-ordinate, then parallel to the Y-axis by the y co-ordinate.

L in Quadrant II, M in Quadrant I, N in Quadrant III, P and Q in Quadrant IV, S on the X-axis, T on the Y-axis

1On 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?

OA = 3 units along the X-axis and OC = 3 units along the Y-axis.

Seg AB is parallel to the Y-axis and seg BC is parallel to the X-axis; AB = BC = 3 units.

So seg AB and seg BC, together with seg OA and seg OC on the axes, form a quadrilateral with all sides 3 units and all angles 90°.

∴ The figure formed is the square OABC of side 3 units

2Write the equation of the line parallel to the Y-axis at a distance of 7 units from it to its left.

A line parallel to the Y-axis has equation x = a. To the left of the Y-axis, a is negative.

∴ x = −7

3Write the equation of the line parallel to the X-axis at a distance of 5 units from it and below the X-axis.

A line parallel to the X-axis has equation y = b. Below the X-axis, b is negative.

∴ y = −5

4The point Q(−3, −2) lies on a line parallel to the Y-axis. Write the equation of the line and draw its graph.

Every point on a line parallel to the Y-axis has the same x co-ordinate. Q has x co-ordinate −3.

∴ The equation of the line is x = −3

Graph of x = −3, passing through Q(−3, −2)

5X-axis and line x = −4 are parallel lines. What is the distance between them?

The line x = −4 is parallel to the Y-axis, not the X-axis (it is perpendicular to the X-axis). The question is meant to read “Y-axis and line x = −4”.

The line x = −4 is 4 units to the left of the Y-axis.

∴ The distance between the Y-axis and the line x = −4 is 4 units

6Which 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 (ii) y − 2 = 0 (iii) x + 6 = 0 (iv) y = −5

(i) x = 3 is of the form x = a.

(ii) y − 2 = 0 means y = 2, of the form y = b.

(iii) x + 6 = 0 means x = −6, of the form x = a.

(iv) y = −5 is of the form y = b.

Parallel to the X-axis: (ii) and (iv)

Parallel to the Y-axis: (i) and (iii)

7On 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.

For each point, x + y = 5: A: 2 + 3 = 5, B: 6 + (−1) = 5, C: 0 + 5 = 5.

So all three points lie on the line x + y = 5; they are collinear.

On the Y-axis x = 0 ∴ y = 5, which is point C(0, 5).

On the X-axis y = 0 ∴ x = 5, giving the point (5, 0).

∴ The line meets the X-axis at (5, 0) and the Y-axis at (0, 5)

8Draw 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

x + 4 = 0 ⇒ x = −4 (parallel to Y-axis)

y − 1 = 0 ⇒ y = 1 (parallel to X-axis)

2x + 3 = 0 ⇒ x = −1.5 (parallel to Y-axis)

3y − 15 = 0 ⇒ y = 5 (parallel to X-axis)

Each vertical line meets each horizontal line; the two vertical lines are parallel, as are the two horizontal lines.

∴ Points of intersection: (−4, 1), (−4, 5), (−1.5, 1), (−1.5, 5)

9Draw the graphs of the equations given below. (i) x + y = 2 (ii) 3x − y = 0 (iii) 2x + y = 1

For each equation, find a few ordered pairs, plot them and draw the line through them.

(i) x + y = 2 ⇒ y = 2 − x

x	0	1	2	-1
y	2	1	0	3
(x, y)	(0, 2)	(1, 1)	(2, 0)	(−1, 3)

(ii) 3x − y = 0 ⇒ y = 3x

x	0	1	-1
y	0	3	−3
(x, y)	(0, 0)	(1, 3)	(−1, −3)

(iii) 2x + y = 1 ⇒ y = 1 − 2x

x	0	1	-1	2
y	1	−1	3	−3
(x, y)	(0, 1)	(1, −1)	(−1, 3)	(2, −3)

1Choose the correct alternative answer for the following questions.

(i) What is the form of co-ordinates of a point on the X-axis? — (C) (a, 0)
The y co-ordinate of every point on the X-axis is 0.
(ii) Any point on the line y = x is of the form — (A) (a, a)
Both co-ordinates are equal.
(iii) What is the equation of the X-axis? — (B) y = 0
Every point on the X-axis has y = 0.
(iv) In which quadrant does the point (−4, −3) lie? — (C) Third
Both co-ordinates are negative.
(v) What is the nature of the line which includes the points (−5, 5), (6, 5), (−3, 5), (0, 5)? — (C) Parallel to X-axis
All points have y = 5, so the line is y = 5.
(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? — (B) Q and R
The fourth quadrant needs x positive and y negative.

2Some points are shown in the figure 7.11. With the help of it answer the following questions.

Fig. 7.11

Q(−2, 2), R(4, −1)
T(0, −1), M(3, 0)
Point S(−3, −2) lies in the third quadrant.
None of the lettered points P, Q, R, S, T, M has equal x and y co-ordinates. Only the origin O(0, 0) has them equal.

3Without plotting the points on a graph, state in which quadrant or on which axis do the following points lie. (i) (5, −3) (ii) (−7, −12) (iii) (−23, 4) (iv) (−9, 5) (v) (0, −3) (vi) (−6, 0)

Point	Quadrant / axis
(i) (5, −3)	Quadrant IV
(ii) (−7, −12)	Quadrant III
(iii) (−23, 4)	Quadrant II
(iv) (−9, 5)	Quadrant II
(v) (0, −3)	Y-axis
(vi) (−6, 0)	X-axis

4Plot 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)

A in Quadrant I, D in Quadrant II, B in Quadrant III, C in Quadrant IV, E on the Y-axis, F on the X-axis

5In 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? (ii) Write the co-ordinates of the points P, Q and R. (iii) What is the difference between the x co-ordinates of the points L and M?

Fig. 7.12

Line LM is x = 3, so its distance from the Y-axis is 3 units.
P(3, 2), Q(3, −1), R(3, 0)
L and M both have x co-ordinate 3, so the difference is 3 − 3 = 0.

6How many lines are there which are parallel to X-axis and having a distance 5 units?

One line is 5 units above the X-axis: y = 5.

One line is 5 units below the X-axis: y = −5.

∴ There are 2 such lines: 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?

The line x = a is parallel to the Y-axis and passes through (a, 0).

If a > 0 it is a units to the right; if a < 0 it is −a units to the left; if a = 0 it is the Y-axis itself.

∴ The distance is |a| units

7. Co-ordinate Geometry

Practice Set 7.1

Practice Set 7.2

Problem Set 7