If three point (h, 0), (a, b) and (0, k) lie on a line, show that \(\frac ah+ \frac bk=1\).

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

Given:

Three points (h, 0), (a, b) and (0, k) lie on the same straight line.

Step 1: Write the intercept form of a line

The equation of a line which cuts the x-axis at (h, 0) and the y-axis at (0, k) is:

x/h + y/k = 1

Step 2: Use the fact that (a, b) lies on this line

Since the point (a, b) lies on the line, it must satisfy its equation.

Substitute x = a and y = b in the equation:

a/h + b/k = 1

Conclusion:

If the three points (h, 0), (a, b) and (0, k) are collinear, then
a/h + b/k = 1

If three point (h, 0), (a, b) and (0, k) lie on a line, show that \(\frac ah+ \frac bk=1\).

Solution and Explanation

Top Questions on Various Forms of the Equation of a Line