Find the equation of the line which intersects the x-axis at a distance of 3 units to the left of origin with slope -2.

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 x-axis at a distance of 3 units to the left of the origin.
Slope of the line = −2

Step 1: Identify a point on the line

3 units to the left of the origin on the x-axis corresponds to the point:

(−3, 0)

Step 2: Use point–slope form of a line

Equation of a line:

y − y₁ = m(x − x₁)

Here,

m = −2, (x₁, y₁) = (−3, 0)

y − 0 = −2(x + 3)

Step 3: Simplify

y = −2x − 6

Final Answer:

The equation of the required line is:
y = −2x − 6

Find the equation of the line which intersects the x-axis at a distance of 3 units to the left of origin with slope -2.

Solution and Explanation

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