Coordinate Geometry Practice Paper - Class 10th Math 2
Introduction to Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of mathematics that studies geometric shapes using coordinates. It connects algebra with geometry through the use of a Cartesian coordinate system.
In this practice paper, we will explore various concepts related to coordinate geometry that are important for Class 10 students preparing for their examinations.
Important Concepts and Formulas
Distance Formula
The distance between two points A(x, y) and B(x, y) is given by:
AB = [(x x) + (y y)]
Section Formula
The coordinates of point P which divides the line segment joining A(x, y) and B(x, y) internally in the ratio m:n are:
P = [(mx + nx)/(m+n), (my + ny)/(m+n)]
Midpoint Formula
The coordinates of the midpoint M of the line segment joining A(x, y) and B(x, y) are:
M = [(x + x)/2, (y + y)/2]
Area of Triangle
The area of a triangle formed by three points A(x, y), B(x, y), and C(x, y) is:
Area = |x(y y) + x(y y) + x(y y)|
Practice Questions
Question 1
Find the distance between the points (3, 4) and (7, 1).
Solution:
Using the distance formula: AB = [(x x) + (y y)]
Given points: A(3, 4) and B(7, 1)
AB = [(7 3) + (1 4)]
AB = [(4) + (-3)]
AB = [16 + 9]
AB = 25
AB = 5 units
Question 2
Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by (1, -2).
Solution:
Let the ratio be k:1
Using section formula:
x-coordinate: 1 = (k 6 + (-3) 1)/(k+1)
1 = (6k - 3)/(k+1)
k + 1 = 6k - 3
5k = 4
k = 4/5
Therefore, the point (1, -2) divides the line segment in the ratio 4:5.
Question 3
Find the coordinates of the midpoint of the line segment joining the points (4, -6) and (2, -8).
Solution:
Using midpoint formula:
M = [(x + x)/2, (y + y)/2]
Given points: A(4, -6) and B(2, -8)
M = [(4 + 2)/2, (-6 + -8)/2]
M = [6/2, -14/2]
M = [3, -7]
Therefore, the midpoint of the line segment is (3, -7).
Question 4
Find the area of the triangle whose vertices are (1, -1), (-4, 6), and (-3, -5).
Solution:
Using the area of triangle formula:
Area = |x(y y) + x(y y) + x(y y)|
Given vertices: A(1, -1), B(-4, 6), and C(-3, -5)
Area = |1(6 (-5)) + (-4)((-5) (-1)) + (-3)(-1 6)|
Area = |1(11) + (-4)(-4) + (-3)(-7)|
Area = |11 + 16 + 21|
Area = |48|
Area = 24 square units
Question 5
Find the value of k if the points A(2, 3), B(4, k), and C(6, -3) are collinear.
Solution:
For collinear points, the area of the triangle formed by them must be zero.
Using the area of triangle formula:
0 = |2(k (-3)) + 4(-3 3) + 6(3 k)|
0 = |2(k + 3) + 4(-6) + 6(3 k)|
0 = |2k + 6 - 24 + 18 - 6k|
0 = |-4k|
0 = -2k
k = 0
Therefore, the value of k is 0.
Question 6
Determine if the points (1, 5), (2, 3), and (-2, -11) are collinear.
Solution:
To check if the points are collinear, we find the area of the triangle formed by them.
Using the area of triangle formula:
Area = |1(3 (-11)) + 2(-11 5) + (-2)(5 3)|
Area = |1(14) + 2(-16) + (-2)(2)|
Area = |14 - 32 - 4|
Area = |-22|
Area = 11
Since the area is not zero, the points are not collinear.
Question 7
Find a relation between x and y such that the point (x, y) is equidistant from the points (3, 6) and (-3, 4).
Solution:
Using the distance formula to equate distances from (x, y) to (3, 6) and (-3, 4):
[(x-3) + (y-6)] = [(x-(-3)) + (y-4)]
[(x-3) + (y-6)] = [(x+3) + (y-4)]
Squaring both sides:
(x-3) + (y-6) = (x+3) + (y-4)
x - 6x + 9 + y - 12y + 36 = x + 6x + 9 + y - 8y + 16
-6x - 12y + 45 = 6x - 8y + 25
-12x - 4y + 20 = 0
3x + y = 5
Therefore, the relation between x and y is 3x + y = 5.
Question 8
Find the coordinates of the points which divide the line segment joining A(4, -3) and B(8, 5) into four equal parts.
Solution:
We need to find points P, Q, and R that divide AB into four equal parts.
First, find the midpoint:
Midpoint = [(4+8)/2, (-3+5)/2] = [6, 1]
This is point Q.
Now, we find point P, which divides the segment from A to Q in the ratio 1:1:
P = [(4+6)/2, (-3+1)/2] = [5, -1]
Next, we find point R, which divides the segment from Q to B in the ratio 1:1:
R = [(6+8)/2, (1+5)/2] = [7, 3]
Therefore, the points are P(5, -1), Q(6, 1), and R(7, 3).
Tips for Solving Coordinate Geometry Problems
- Always draw a diagram to visualize the problem.
- Clearly write down the given information.
- Identify which formula is most appropriate for the question.
- Show all steps of your calculation.
- Check your final answer to ensure it makes sense in the context of the problem.
- Practice regularly to improve speed and accuracy.
Note for Students
Coordinate geometry is an important topic that requires conceptual understanding and regular practice. Make sure to learn the formulas and understand their derivations. This will help you apply them correctly in complex problems.
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