SUMMARY: The chapter on Coordinate Geometry introduces the Cartesian plane and the concepts of plotting points using coordinates. KEY TOPICS: Cartesian plane, x-axis, y-axis, origin, coordinates, quadrants, plotting points, distance formula, section formula, applications of coordinate geometry
What is the coordinate of the origin in the Cartesian plane?
A(0, 0)
B(1, 1)
C(0, 1)
D(1, 0)
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Correct answer: Option 1 — (0, 0)
Q21 Mark
In which quadrant would the point (-3, 5) be located?
AQuadrant I
BQuadrant II
CQuadrant III
DQuadrant IV
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Correct answer: Option 2 — Quadrant II
Q31 Mark
If point A has coordinates (2, 3) and point B has coordinates (5, 7), what is the distance between points A and B?
A5 units
B4 units
C3 units
D√13 units
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Correct answer: Option 4 — √13 units
Q41 Mark
Which of the following points lies on the x-axis?
A(4, 0)
B(0, 5)
C(3, 2)
D(0, 0)
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Correct answer: Option 1 — (4, 0)
Q51 Mark
Using the section formula, what is the coordinate of the point that divides the line segment joining (2, 3) and (4, 7) in the ratio 1:2?
A(3, 5)
B(2.67, 4.67)
C(3.33, 5.33)
D(3, 6)
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Correct answer: Option 3 — (3.33, 5.33)
Short Answer Questions5 questions
Q63 Marks
What are the coordinates of the origin in the Cartesian plane?
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The coordinates of the origin in the Cartesian plane are (0, 0). This point is where the x-axis and y-axis intersect.
Q73 Marks
Explain the significance of the four quadrants in the Cartesian plane.
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The four quadrants in the Cartesian plane are defined by the signs of the coordinates. Quadrant I has positive x and y coordinates, Quadrant II has negative x and positive y, Quadrant III has negative x and y, and Quadrant IV has positive x and negative y.
Q83 Marks
How do you plot the point (3, -2) on the Cartesian plane?
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To plot the point (3, -2), start at the origin (0, 0). Move 3 units to the right along the x-axis and then 2 units down along the y-axis, marking the point where you land.
Q93 Marks
What is the distance formula used for in coordinate geometry?
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The distance formula is used to calculate the distance between two points in the Cartesian plane. If the points are (x1, y1) and (x2, y2), the distance d is given by d = √((x2 - x1)² + (y2 - y1)²).
Q103 Marks
Using the section formula, find the coordinates of the point that divides the line segment joining (2, 3) and (4, 7) in the ratio 1:2.
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Using the section formula, the coordinates of the point dividing the segment in the ratio 1:2 can be calculated as follows: x = (1*4 + 2*2)/(1+2) = 8/3 and y = (1*7 + 2*3)/(1+2) = 13/3. Thus, the coordinates are (8/3, 13/3).
Long Answer Questions5 questions
Q116 Marks
Explain the significance of the Cartesian plane in coordinate geometry. How are the x-axis and y-axis defined, and what role does the origin play in this system?
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The Cartesian plane is a two-dimensional plane formed by the intersection of two perpendicular lines, known as the x-axis and y-axis. The x-axis runs horizontally, while the y-axis runs vertically. The point where these two axes intersect is called the origin, denoted as (0,0). The Cartesian plane allows for the representation of points using ordered pairs of numbers, which correspond to their positions along the x and y axes. This system is fundamental in coordinate geometry as it provides a framework for plotting points, analyzing geometric shapes, and solving problems involving distance and area.
Q126 Marks
Describe the four quadrants of the Cartesian plane. How are the coordinates of points in each quadrant determined?
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The Cartesian plane is divided into four quadrants based on the signs of the x and y coordinates. Quadrant I is located in the upper right, where both x and y coordinates are positive. Quadrant II is in the upper left, where x is negative and y is positive. Quadrant III is in the lower left, where both coordinates are negative. Finally, Quadrant IV is in the lower right, where x is positive and y is negative. The determination of a point's quadrant is based on the signs of its coordinates; for example, a point with coordinates (-3, 4) lies in Quadrant II, while (5, -2) lies in Quadrant IV.
Q136 Marks
Using the distance formula, calculate the distance between the points A(2, 3) and B(5, 7). Show your working and explain each step.
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To calculate the distance between points A(2, 3) and B(5, 7), we use the distance formula, which is given by D = √[(x2 - x1)² + (y2 - y1)²]. Here, (x1, y1) = (2, 3) and (x2, y2) = (5, 7). First, we find the differences: (x2 - x1) = 5 - 2 = 3 and (y2 - y1) = 7 - 3 = 4. Next, we square these differences: 3² = 9 and 4² = 16. Adding these gives us 9 + 16 = 25. Finally, we take the square root: D = √25 = 5. Therefore, the distance between points A and B is 5 units.
Q146 Marks
Explain the section formula in coordinate geometry. How can it be used to find the coordinates of a point that divides the line segment joining points A(1, 2) and B(4, 6) in the ratio 2:3?
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The section formula is used to find the coordinates of a point that divides a line segment into a specific ratio. If point P divides the line segment AB, where A(x1, y1) and B(x2, y2) are the endpoints, in the ratio m:n, the coordinates of point P can be calculated using the formula P(x, y) = ((mx2 + nx1) / (m+n), (my2 + ny1) / (m+n)). For points A(1, 2) and B(4, 6) dividing in the ratio 2:3, we set m = 2 and n = 3. Thus, the coordinates of P are calculated as follows: x = (2*4 + 3*1) / (2+3) = (8 + 3) / 5 = 11/5 = 2.2 and y = (2*6 + 3*2) / (2+3) = (12 + 6) / 5 = 18/5 = 3.6. Therefore, the coordinates of point P are (2.2, 3.6).
Q156 Marks
Plot the points A(3, 4), B(-2, 5), and C(-1, -3) on the Cartesian plane. Describe the position of each point in relation to the origin and the axes.
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To plot the points A(3, 4), B(-2, 5), and C(-1, -3) on the Cartesian plane, we start by identifying their coordinates. Point A(3, 4) is located in Quadrant I, 3 units to the right of the origin along the x-axis and 4 units up along the y-axis. Point B(-2, 5) is in Quadrant II, 2 units to the left of the origin and 5 units up. Point C(-1, -3) is in Quadrant III, 1 unit to the left of the origin and 3 units down. Each point's position relative to the origin and the axes highlights its quadrant placement and the signs of its coordinates, demonstrating the relationship between the points and the Cartesian plane.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The coordinates of the origin in the Cartesian plane are (0, 0).
Reason (R): The origin is the point where the x-axis and y-axis intersect.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): In the Cartesian plane, the point (3, -2) lies in the second quadrant.
Reason (R): The second quadrant contains points where the x-coordinate is negative and the y-coordinate is positive.
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Correct answer: Option 4 —
A is false, but R is true.
Q181 Mark
Assertion (A): The distance formula can be used to find the distance between two points in the Cartesian plane.
Reason (R): The distance formula is derived from the Pythagorean theorem.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q191 Mark
Assertion (A): The coordinates of a point in the first quadrant are always positive.
Reason (R): In the first quadrant, both x and y coordinates are negative.
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Correct answer: Option 3 —
A is true, but R is false.
Q201 Mark
Assertion (A): The section formula is used to find the coordinates of a point dividing a line segment in a given ratio.
Reason (R): The section formula calculates the coordinates based on the weighted average of the endpoints.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The x-axis is the horizontal line in the Cartesian plane.
Statement 2: The y-axis is the vertical line in the Cartesian plane.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: The coordinates of the origin are (0, 0).
Statement 2: In the first quadrant, both x and y coordinates are negative.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q231 Mark
Statement 1: The distance formula can be used to find the distance between two points in the Cartesian plane.
Statement 2: The section formula is used to find the coordinates of a point that divides a line segment in a given ratio.
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Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: The point (3, -2) lies in the second quadrant.
Statement 2: The coordinates of a point are always written in the form (y, x).
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Correct answer: Option 4 —
Both statements are false.
Q251 Mark
Statement 1: In the Cartesian plane, the first quadrant is where both x and y coordinates are positive.
Statement 2: The coordinates of a point are represented as (x, y).
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Correct answer: Option 1 —
Both statements are true.
