NCERT Class X Chapter 7: Coordinate Geometry Example 3

NCERT Class X Chapter 7: Coordinate Geometry Example 3

Question:

Ashima, Bharti and Camella are seated at A(3, 1), B(6, 4) and C(8, 6) respectively. Do you think they are seated in a line? Give reasons for your answer.

Given:

Coordinates of Ashima (A): (3, 1)
Coordinates of Bharti (B): (6, 4)
Coordinates of Camella (C): (8, 6)

To Find:

Whether the points A, B, and C are collinear.

Formula:

The slope of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ If the slopes between all pairs of points are equal, the points are collinear.

Solution:

Step 1: Find the slope of AB.

$$ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{6 - 3} = \frac{3}{3} = 1 $$

Step 2: Find the slope of BC.

$$ m_{BC} = \frac{6 - 4}{8 - 6} = \frac{2}{2} = 1 $$

Step 3: Compare the slopes.

$$ m_{AB} = m_{BC} = 1 $$
Since the slopes are equal, the points A, B, and C are collinear.

Result:

Yes, Ashima, Bharti, and Camella are seated in a line because the slope between each pair of points is the same.

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