NCERT Class X Chapter 7: Coordinate Geometry Exercise 7.1 Question 3

NCERT Class X Chapter 7: Coordinate Geometry

Question:

Determine if the points (1, 5), (2, 3) and (–2, –11) are collinear.

Given:

Points: \( A(1, 5) \), \( B(2, 3) \), \( C(-2, -11) \)

To Find:

Are the points \( A \), \( B \), and \( C \) collinear?

Formula:

Area of triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), \( (x_3, y_3) \):
$$ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| $$ If the area is 0, the points are collinear.

Solution:

Step 1: Assign coordinates to the points.

$$ A(x_1, y_1) = (1, 5),\quad B(x_2, y_2) = (2, 3),\quad C(x_3, y_3) = (-2, -11) $$

Step 2: Substitute the values in the area formula.

$$ \text{Area} = \frac{1}{2} \left| 1(3 - (-11)) + 2((-11) - 5) + (-2)(5 - 3) \right| $$

Step 3: Simplify the expressions inside the modulus.

$$ = \frac{1}{2} \left| 1 \times 14 + 2 \times (-16) + (-2) \times 2 \right| $$

Step 4: Calculate each term.

$$ = \frac{1}{2} \left| 14 - 32 - 4 \right| $$

Step 5: Add the values inside the modulus.

$$ = \frac{1}{2} \left| -22 \right| $$

Step 6: Find the area.

$$ = \frac{1}{2} \times 22 = 11 $$

Step 7: Since the area is not zero, the points are not collinear.

Result:

The points (1, 5), (2, 3), and (–2, –11) are not collinear.

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