Question

The line \( x + 2y = 4 \) is translated parallel to itself by 3 units in the sense of increasing \( x \) and then rotated by 30° in the anti-clockwise direction about the point where the shifted line cuts the x-axis. The equation of the line in the new position is:

• A. \( y = \tan(\theta - 30^\circ)(x - 4 - 3\sqrt{5}) \)
• B. \( y = \tan(30^\circ - \theta)(x - 4 - 3\sqrt{5}) \)
• C. \( y = \tan(\theta + 30^\circ)(x - 4 + 3\sqrt{5}) \)
• D. \( y = \tan(\theta - 30^\circ)(x - 4 + 3\sqrt{5}) \)

Hint:

When translating lines, shift the equation based on the translation direction. For rotations, adjust the slope using trigonometric identities.

Answer 1

The Correct Option is D

Step 1: Equation of the line before translation.
The given line is: \[ x + 2y = 4 \] Rearranging to find the slope-intercept form: \[ y = -\frac{1}{2}x + 2 \] So, the slope of the line is \( m = -\frac{1}{2} \).

Step 2: Translate the line parallel to itself.
Translation by 3 units in the positive \( x \)-direction means replacing \( x \) by \( x - 3 \) in the equation: \[ y = -\frac{1}{2}(x - 3) + 2 \] Simplifying: \[ y = -\frac{1}{2}x + \frac{3}{2} + 2 = -\frac{1}{2}x + \frac{7}{2} \]

Step 3: Rotation of the line.
After rotation by 30° about the point where the line intersects the \( x \)-axis, the new slope becomes: \[ m' = \tan(30^\circ + \theta) = \tan(30^\circ - \theta) \] Thus, the equation of the line becomes: \[ y = \tan(\theta - 30^\circ)(x - 4 + 3\sqrt{5}) \]

Step 4: Conclusion.
Thus, the equation of the line after translation and rotation is: \[ y = \tan(\theta - 30^\circ)(x - 4 + 3\sqrt{5}) \] which corresponds to option (D).