RS Aggarwal Class 9 Mathematics Solutions Chapter 6 Coordinate Geometry

 

Question 1. Draw the perpendiculars from AF, BG, CH, DI and EJ on the x-axis.
Answer: By drawing perpendiculars from the given points to the x-axis, we can determine the coordinates of each point:
(1) The perpendicular distance from A to the y-axis is OF = -6 units, and the perpendicular distance from A to the x-axis is AF = 5 units. Therefore, A has coordinates (-6, 5).
(2) The perpendicular distance from B to the y-axis is OG = 5 units, and the perpendicular distance from B to the x-axis is BG = 4 units. Therefore, B has coordinates (5, 4).
(3) The perpendicular distance from C to the y-axis is OH = -3 units, and the perpendicular distance from C to the x-axis is HC = 2 units. Therefore, C has coordinates (-3, 2).
(4) The perpendicular distance from D to the y-axis is OI = 2 units, and the perpendicular distance from D to the x-axis is ID = -2 units. Therefore, D has coordinates (2, -2).
(5) The perpendicular distance from E to the y-axis is OJ = -1 unit, and the perpendicular distance from E to the x-axis is JE = -4 units. Therefore, E has coordinates (-1, -4).
The coordinates of points A, B, C, D and E are respectively A(-6, 5), B(5, 4), C(-3, 2), D(2, -2) and E(-1, -4).
In simple words: To find a point's coordinates, measure how far it is from the y-axis (this gives the first number) and how far it is from the x-axis (this gives the second number). Include the sign based on which side of the axis the point lies.

Exam Tip: Always remember the order: x-coordinate (horizontal distance) comes first, followed by y-coordinate (vertical distance). Use negative signs for distances on the left of the y-axis or below the x-axis.

 

Question 2. Let X'OX and Y'OY be the coordinate axes. Fix the side of the small squares as one unit.
Answer: Following the given instructions with each small square having a side length of one unit:
(i) From the origin O, move 7 units in the positive x-direction and then 4 units in the positive y-direction to mark point P(7, 4).
(ii) From the origin O, move 5 units in the negative x-direction and then 3 units in the positive y-direction to mark point Q(-5, 3).
(iii) From the origin O, move 6 units in the negative x-direction and then 3 units in the negative y-direction to mark point R(-6, -3).
(iv) From the origin O, move 3 units in the positive x-direction and then 7 units in the negative y-direction to mark point S(3, -7).
(v) From the origin O, move 6 units in the positive x-direction to mark point A(6, 0).
(vi) From the origin O, move 9 units in the positive y-direction to mark point B(0, 9).
(vii) Mark the origin as O(0, 0).
(viii) From the origin O, move 3 units in the negative x-direction and then 3 units in the negative y-direction to mark point C(-3, -3).
These points are shown in the accompanying graph.
In simple words: To plot a point like (7, 4), start at the origin, go 7 units right, then go 4 units up. For negative numbers, go left for x and down for y. Each unit on the graph represents one small square.

Exam Tip: Use a ruler for accuracy when marking points on graph paper. Always plot the x-coordinate first (along the horizontal axis) before moving in the y-direction (along the vertical axis).

 

Question 3. Identify which axis each of the following points lies on.
Answer:
(i) In the coordinate pair (7, 0), the y-coordinate equals 0. When the y-coordinate is zero, the point must lie on the x-axis. So (7, 0) lies on the x-axis.
(ii) In the coordinate pair (0, -5), the x-coordinate equals 0. When the x-coordinate is zero, the point must lie on the y-axis. So (0, -5) lies on the y-axis.
(iii) In the coordinate pair (0, 1), the x-coordinate equals 0. When the x-coordinate is zero, the point must lie on the y-axis. So (0, 1) lies on the y-axis.
(iv) In the coordinate pair (-4, 0), the y-coordinate equals 0. When the y-coordinate is zero, the point must lie on the x-axis. So (-4, 0) lies on the x-axis.
In simple words: If the second number (y-coordinate) is 0, the point is on the x-axis. If the first number (x-coordinate) is 0, the point is on the y-axis.

Exam Tip: Any point on an axis will always have one coordinate equal to zero. This is a quick way to check if your coordinate identification is correct.

 

Question 4. Identify which quadrant each of the following points lies in: (i) (-6, 5) (ii) (-3, -2) (iii) (2, -9)
Answer:
(i) Points with the form (negative, positive) are located in the second quadrant. Since (-6, 5) has a negative x-coordinate and a positive y-coordinate, the point (-6, 5) lies in Quadrant II.
(ii) Points with the form (negative, negative) are located in the third quadrant. Since (-3, -2) has both coordinates negative, the point (-3, -2) lies in Quadrant III.
(iii) Points with the form (positive, negative) are located in the fourth quadrant. Since (2, -9) has a positive x-coordinate and a negative y-coordinate, the point (2, -9) lies in Quadrant IV.
In simple words: Remember the pattern: top-left is Quadrant II (negative, positive), bottom-left is Quadrant III (negative, negative), and bottom-right is Quadrant IV (positive, negative). The top-right is Quadrant I (positive, positive).

Exam Tip: A quick way to remember the quadrants is to think of them going counter-clockwise starting from the top-right: I (top-right), II (top-left), III (bottom-left), IV (bottom-right).

 

Question 5. Draw the graph of the equation y = x + 1.
Answer: To create the graph of y = x + 1, we first find several points that satisfy the equation:
When x = 1: y = 1 + 1 = 2
When x = 2: y = 2 + 1 = 3

On graph paper, draw the coordinate axes X'OX and Y'OY to represent the x-axis and y-axis. Then plot the points P(1, 2) and Q(2, 3) on the graph. Connect these two points with a straight line and extend it in both directions. The line PQ represents the complete graph of the equation y = x + 1.
In simple words: Pick any two x-values, find the matching y-values using the equation, plot those points, and draw a line through them. The line will continue forever in both directions.

Exam Tip: Always extend the line in both directions to show that it continues infinitely. Use at least two points to ensure your line is drawn accurately.

 

Question 6. Draw the graph of the equation y = 3x + 2.
Answer: To create the graph of y = 3x + 2, we first calculate points that satisfy the equation:
When x = 1: y = (3 × 1) + 2 = 5
When x = 2: y = (3 × 2) + 2 = 8

On graph paper, draw the axes X'OX and Y'OY as the x-axis and y-axis respectively. Plot points P(1, 5) and Q(2, 8) on the paper. Join these points with a line and extend it across both sides. The resulting line PQ represents the graph of y = 3x + 2.
In simple words: Substitute x-values into the equation to find y-values, mark those points on the graph, draw a line through them, and extend it in both directions.

Exam Tip: Check that your plotted points actually satisfy the equation before drawing the line. This prevents errors from affecting your final graph.

 

Question 7. Draw the graph of the equation y = 5x - 3.
Answer: To create the graph of y = 5x - 3, we calculate points that satisfy the equation:
When x = 0: y = (5 × 0) - 3 = -3
When x = 1: y = (5 × 1) - 3 = 2

On graph paper, draw the axes X'OX and Y'OY as the x-axis and y-axis respectively. Plot points P(0, -3) and Q(1, 2) on the paper. Connect these two points and extend the line in both directions. The resulting line PQ shows the graph of y = 5x - 3.
In simple words: Choose simple x-values, work out the corresponding y-values, mark those points, draw a straight line through them, and extend it both ways.

Exam Tip: Using x = 0 is often helpful because it directly gives you the y-intercept. This point is useful for drawing your line accurately.

 

Question 8. Draw the graph of the equation y = 3x.
Answer: To create the graph of y = 3x, we calculate points that satisfy the equation:
When x = 1: y = 3 × 1 = 3
When x = 2: y = 3 × 2 = 6

On graph paper, draw the axes X'OX and Y'OY as the x-axis and y-axis respectively. Plot points P(1, 3) and Q(2, 6) on the paper. Connect these points with a straight line and extend it in both directions. The resulting line PQ shows the graph of y = 3x.
In simple words: For this equation, when you multiply x by 3, you get y. Pick x-values, multiply by 3, plot those points, and draw a line through them.

Exam Tip: Notice that this line passes through the origin (0, 0) because when x = 0, y = 0. This is a helpful check for lines of the form y = mx.

 

Question 9. Draw the graph of the equation y = -x.
Answer: To create the graph of y = -x, we calculate points that satisfy the equation:
When x = 1: y = -1
When x = 2: y = -2

On graph paper, draw the axes X'OX and Y'OY as the x-axis and y-axis respectively. Plot points P(1, -1) and Q(2, -2) on the paper. Connect these points with a line and extend it in both directions. The resulting line PQ represents the graph of y = -x.
In simple words: For each x-value, the y-value is the opposite sign. So positive x gives negative y, and vice versa. The line slopes downward from left to right.

Exam Tip: This line passes through the origin and has a negative slope, meaning it goes downward as you move from left to right. Use this visual check to confirm your graph is correct.