OP Malhotra Class 11 Maths Solutions Chapter 15 Basic Concepts of Points and their Coordinates Exercise 15 (A)

S Chand Class 11 ICSE Maths Solutions Chapter 15 Basic Concepts of Points and their Coordinates Ex 15(a)

Question 1. Find the coordinates of the points shown on the graph in Fig. given below :
Answer:
(i) Point P lies on the x-axis, 3 units to the left of the origin (O). So, its coordinates are \( (-3, 0) \).
(ii) Point Q is 2 units to the right of the origin along the positive x-axis, and then 5 units up along the positive y-axis. So, its coordinates are \( (2, 5) \).
(iii) Point O is the origin itself, where the x-axis and y-axis meet. Its coordinates are \( (0, 0) \).
(iv) Point R is 3 units to the left of the origin along the negative x-axis, and then 5 units down along the negative y-axis. So, its coordinates are \( (-3, -5) \).
(v) Point S lies on the x-axis, 3 units to the right of the origin (O). So, its coordinates are \( (3, 0) \).
(vi) Point A is 4 units to the right of the origin along the positive x-axis, and then 3 units down along the negative y-axis. So, its coordinates are \( (4, -3) \).
(vii) Point L is 4 units to the left of the origin along the negative x-axis, and then 2 units down along the negative y-axis. So, its coordinates are \( (-4, -2) \).
(viii) Point M lies on the y-axis, 2 units up from the origin along the positive y-axis. So, its coordinates are \( (0, 2) \).In simple words: To find the coordinates of a point, first see how far it is from the center (origin) left or right (x-value), then how far it is up or down (y-value). A positive number means right or up, and a negative number means left or down.

🎯 Exam Tip: Always remember to write the x-coordinate first, followed by the y-coordinate, inside parentheses like \( (x, y) \). The x-axis is horizontal and the y-axis is vertical.

Question 2. Plot the points (2, 3),(-5, -7),(-4, 6), (0, 8),(-5, 0) on the graph paper.
Answer:
The location of each point is determined by its x and y coordinates:
(i) Point \( P(2, 3) \) is in the first quadrant. To plot it, move 2 units right from the origin along the x-axis, then 3 units up along the y-axis.
(ii) Point \( T(-5, 0) \) is on the negative x-axis. To plot it, move 5 units left from the origin along the x-axis. Since the y-coordinate is 0, it stays on the x-axis.
(iii) Point \( R(-4, 6) \) is in the second quadrant. To plot it, move 4 units left from the origin along the x-axis, then 6 units up along the y-axis.
(iv) Point \( S(0, 8) \) is on the positive y-axis. To plot it, move 8 units up from the origin along the y-axis. Since the x-coordinate is 0, it stays on the y-axis.
(v) Point \( Q(-5, -7) \) is in the third quadrant. To plot it, move 5 units left from the origin along the x-axis, then 7 units down along the y-axis.In simple words: To put a point on a graph, always start at the middle (the origin). First, move left or right based on the first number (x). Then, move up or down based on the second number (y).

🎯 Exam Tip: When plotting points, always begin at the origin \( (0,0) \). The x-coordinate tells you horizontal movement (left/right) and the y-coordinate tells you vertical movement (up/down). Double-check the signs to move in the correct direction.

Question 3. Where will a point lie if (i) its ordinate is zero, (ii) its abscissa is zero?
Answer:
(i) If a point's ordinate (the y-coordinate) is zero, it means the point has no vertical distance from the x-axis. Therefore, such a point will always lie on the x-axis. For example, points like \( (5, 0) \) or \( (-2, 0) \) are on the x-axis.
(ii) If a point's abscissa (the x-coordinate) is zero, it means the point has no horizontal distance from the y-axis. Therefore, such a point will always lie on the y-axis. For example, points like \( (0, 7) \) or \( (0, -3) \) are on the y-axis.In simple words: If the 'up or down' number (y-coordinate) is zero, the point is on the horizontal line (x-axis). If the 'left or right' number (x-coordinate) is zero, the point is on the vertical line (y-axis).

🎯 Exam Tip: Remember these fundamental definitions: abscissa refers to the x-coordinate and ordinate refers to the y-coordinate. This helps correctly identify a point's position relative to the axes.

Question 4. Where will a point lie if (i) the abscissa equals the ordinate; (ii) the positive abscissa equals the negative of the positive ordinate?
Answer:
(i) When the abscissa (x-coordinate) equals the ordinate (y-coordinate), it means \( x = y \). All points where \( x = y \) lie on a straight line that passes through the origin and bisects the first and third quadrants. This line makes a 45-degree angle with the positive x-axis and the positive y-axis.
(ii) When the positive abscissa equals the negative of the positive ordinate, it means \( x = -y \), where \( x > 0 \) and \( y > 0 \) initially. This implies that the x-coordinate is positive while the y-coordinate is negative. All such points \( (x, -x) \) lie on a straight line that passes through the origin and bisects the second and fourth quadrants. This line also makes a 45-degree angle with the axes.In simple words: If the two numbers for a point are the same (like \( (2,2) \) or \( (-3,-3) \)), the point will be on the line that cuts across the graph from bottom-left to top-right. If the numbers are opposite (like \( (2,-2) \) or \( (-3,3) \)), the point will be on the line that cuts across from top-left to bottom-right.

🎯 Exam Tip: Understanding the lines \( y=x \) and \( y=-x \) is key for coordinate geometry. Remember that \( y=x \) passes through quadrants I and III, while \( y=-x \) passes through quadrants II and IV.