Find the equation of the line which intersects the y-axis at a distance of 2 units above the origin and makes an angle of 30° with the positive direction of the x-axis.

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:

The line intersects the y-axis at a point 2 units above the origin.
The line makes an angle of 30° with the positive x-axis.

Step 1: Find the slope of the line

Slope of a line making an angle θ with the positive x-axis is:

m = tan θ

Here,

θ = 30°

m = tan 30° = 1/√3

Step 2: Identify the y-intercept

The line cuts the y-axis 2 units above the origin.

So, y-intercept = 2

Step 3: Use slope–intercept form

The equation of a line in slope–intercept form is:

y = mx + c

Substitute m = 1/√3 and c = 2

y = (1/√3)x + 2

Final Answer:

The equation of the required line is:
y = (1/√3)x + 2

Find the equation of the line which intersects the y-axis at a distance of 2 units above the origin and makes an angle of 30° with the positive direction of the x-axis.

Solution and Explanation

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