By using the concept of equation of a line, prove that the three points (3, 0), (– 2, – 2) and (8, 2) are collinear.

Question

If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively are \( \frac{16}{11} (3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{11} (-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

Answer 1

By using the concept of equation of a line, prove that the three points
A(3, 0), B(–2, –2) and C(8, 2) are collinear.

Step 1: Find the equation of the line passing through A(3, 0) and B(–2, –2)

Slope of AB: \( m = \frac{-2 - 0}{-2 - 3} = \frac{-2}{-5} = \frac{2}{5} \)

Equation of the line passing through A(3, 0) with slope \( \frac{2}{5} \):

\( y - 0 = \frac{2}{5}(x - 3) \)

\( 5y = 2x - 6 \)

Step 2: Check whether point C(8, 2) satisfies this equation

Substitute x = 8 and y = 2:

\( 5(2) = 2(8) - 6 \)

\( 10 = 16 - 6 = 10 \)

Since the coordinates of point C satisfy the equation of the line, the three points lie on the same straight line.

Final Conclusion:
The points (3, 0), (–2, –2) and (8, 2) are collinear.

By using the concept of equation of a line, prove that the three points (3, 0), (– 2, – 2) and (8, 2) are collinear.

Solution and Explanation

Top Questions on Various Forms of the Equation of a Line