SUMMARY: The chapter on Coordinate Geometry in Class 10 Mathematics focuses on the study of the Cartesian plane and the application of coordinate systems to solve geometric problems. KEY TOPICS: Cartesian plane, coordinates of a point, distance formula, section formula, area of a triangle, collinearity of points, plotting points, applications of coordinate geometry
What is the distance between the points (3, 4) and (7, 1)?
A5 units
B4 units
C3 units
D6 units
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Correct answer: Option 1 — 5 units
Q21 Mark
Which of the following points lies on the line represented by the equation y = 2x + 3?
A(1, 5)
B(2, 6)
C(0, 3)
D(3, 8)
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Correct answer: Option 3 — (0, 3)
Q31 Mark
If the midpoint of a line segment is (4, -2) and one endpoint is (2, 1), what is the other endpoint?
A(6, -5)
B(8, -3)
C(0, -5)
D(4, -5)
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Correct answer: Option 1 — (6, -5)
Q41 Mark
What is the slope of the line passing through the points (2, 3) and (4, 7)?
A2
B1
C0.5
D4
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Correct answer: Option 1 — 2
Q51 Mark
The equation of a line is given as 3x - 4y + 12 = 0. What is the y-intercept of this line?
A3
B4
C-3
D-4
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Correct answer: Option 3 — -3
Short Answer Questions5 questions
Q63 Marks
What is the distance between the points (3, 4) and (7, 1)?
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To find the distance between two points (x1, y1) and (x2, y2), we use the formula: \(d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\). Substituting the values, we get \(d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\).
Q73 Marks
Find the midpoint of the line segment joining the points (2, 3) and (8, 7).
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The midpoint M of a line segment joining points (x1, y1) and (x2, y2) is given by the formula: \(M = (\frac{x1 + x2}{2}, \frac{y1 + y2}{2})\). Thus, the midpoint is \(M = (\frac{2 + 8}{2}, \frac{3 + 7}{2}) = (5, 5)\).
Q83 Marks
If the point A(2, 3) is reflected in the line y = x, what are the coordinates of the reflected point A'?
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When a point (x, y) is reflected in the line y = x, the coordinates of the reflected point become (y, x). Therefore, the reflected point A' of A(2, 3) is A'(3, 2).
Q93 Marks
Determine the area of the triangle formed by the points (1, 2), (4, 6), and (5, 2).
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The area of a triangle formed by three points (x1, y1), (x2, y2), and (x3, y3) can be calculated using the formula: \(Area = \frac{1}{2} |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|\). Substituting the points, we find the area to be 6 square units.
Q103 Marks
Prove that the points (1, 2), (3, 4), and (5, 6) are collinear.
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To prove that three points are collinear, we can check if the area of the triangle they form is zero. Using the area formula, if the area calculated is zero, the points are collinear. For these points, the area is zero, thus confirming they are collinear.
Long Answer Questions5 questions
Q116 Marks
Explain the concept of the distance formula in coordinate geometry. Derive the distance formula between two points (x1, y1) and (x2, y2) and provide an example to illustrate its application.
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The distance formula is derived from the Pythagorean theorem and is used to calculate the distance between two points in a Cartesian plane. The formula is given by d = √((x2 - x1)² + (y2 - y1)²). For example, to find the distance between the points (3, 4) and (7, 1), we substitute the coordinates into the formula: d = √((7 - 3)² + (1 - 4)²) = √(16 + 9) = √25 = 5. This shows that the distance between the two points is 5 units.
Q126 Marks
Define the section formula in coordinate geometry. Derive the formula for a point dividing the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n. Provide an example to demonstrate its use.
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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 joining points A(x1, y1) and B(x2, y2) 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 instance, if A(2, 3) and B(4, 5) are divided in the ratio 1:2, then P = ((1*4 + 2*2)/(1+2), (1*5 + 2*3)/(1+2)) = (8/3, 11/3).
Q136 Marks
Discuss the concept of the midpoint of a line segment in coordinate geometry. Derive the formula for the midpoint of a line segment joining two points A(x1, y1) and B(x2, y2) and illustrate with an example.
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The midpoint of a line segment is the point that divides the segment into two equal parts. The coordinates of the midpoint M of the line segment joining points A(x1, y1) and B(x2, y2) can be calculated using the formula M = ((x1 + x2)/2, (y1 + y2)/2). For example, if A(1, 2) and B(5, 6), then the midpoint M = ((1 + 5)/2, (2 + 6)/2) = (3, 4). This indicates that the point (3, 4) is equidistant from both A and B.
Q146 Marks
Explain how to find the area of a triangle formed by three points in the coordinate plane. Derive the formula for the area of triangle formed by points A(x1, y1), B(x2, y2), and C(x3, y3) and provide an example.
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The area of a triangle formed by three points in the coordinate plane can be calculated using the formula Area = 1/2 | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |. For example, for points A(1, 2), B(4, 5), and C(7, 2), the area can be calculated as Area = 1/2 | 1(5 - 2) + 4(2 - 2) + 7(2 - 5) | = 1/2 | 1*3 + 0 - 21 | = 1/2 | -18 | = 9 square units.
Q156 Marks
Describe the concept of the slope of a line in coordinate geometry. Derive the formula for the slope of a line passing through two points (x1, y1) and (x2, y2) and provide an example to illustrate your explanation.
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The slope of a line measures its steepness and direction, defined as the change in y over the change in x between two points on the line. The formula for the slope (m) of a line passing through points (x1, y1) and (x2, y2) is given by m = (y2 - y1) / (x2 - x1). For instance, if we take the points (2, 3) and (5, 11), the slope would be m = (11 - 3) / (5 - 2) = 8 / 3. This indicates that for every 3 units moved horizontally, the line rises 8 units vertically.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The distance between the points (3, 4) and (7, 1) can be calculated using the distance formula.
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.
Q171 Mark
Assertion (A): The midpoint of the line segment joining the points (2, 3) and (4, 7) is (3, 5).
Reason (R): The midpoint is calculated by averaging the x-coordinates and y-coordinates 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.
Q181 Mark
Assertion (A): The slope of the line passing through the points (1, 2) and (3, 4) is 1.
Reason (R): The slope is calculated as the change in y divided by the change in x.
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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 the centroid of a triangle are the average of the coordinates of its vertices.
Reason (R): The centroid divides each median in the ratio 2:1.
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Correct answer: Option 2 —
Both A and R are true, but R is not the correct explanation of A.
Q201 Mark
Assertion (A): A line with an undefined slope is vertical.
Reason (R): Vertical lines have a slope of zero.
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Correct answer: Option 3 —
A is true, but R is false.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The coordinates of the midpoint of the line segment joining the points (2, 3) and (4, 7) are (3, 5).
Statement 2: The distance between the points (1, 2) and (4, 6) is 5 units.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: The slope of the line passing through the points (1, 2) and (3, 4) is 1.
Statement 2: The equation of a line with slope 2 passing through the origin is y = 2x + 1.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q231 Mark
Statement 1: The area of a triangle formed by the points (0, 0), (4, 0), and (0, 3) is 6 square units.
Statement 2: The coordinates of the centroid of a triangle with vertices at (2, 3), (4, 5), and (6, 7) are (4, 5).
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Correct answer: Option 2 —
Only Statement 1 is true.
Q241 Mark
Statement 1: The distance formula is derived from the Pythagorean theorem.
Statement 2: The coordinates of the orthocenter of a triangle can be found using the midpoints of its sides.
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Correct answer: Option 4 —
Both statements are false.
Q251 Mark
Statement 1: If two points have the same x-coordinate, the line passing through them is vertical.
Statement 2: The equation of a line parallel to the x-axis is of the form y = mx + c where m = 0.
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Correct answer: Option 1 —
Both statements are true.
