Question

A line passing through the point $A(9, 0)$ makes an angle of $30^\circ$ with the positive direction of the x-axis. If this line is rotated about $A$ through an angle of $15^\circ$ in the clockwise direction, then its equation in the new position is:

• A. \frac{x}{\sqrt{3} + 2} + y = 9 \\\
• B. $\frac{y}{\sqrt{3} - 2} + x = 9$
• C. \frac{x}{\sqrt{3} + 2} + y = 9 \\\
• D. $\frac{x}{\sqrt{3} - 2} + y = 9$

Answer 1

Answer: (B)

$\frac{y}{\sqrt{3} - 2} + x = 9$

Answer 2

The line initially makes an angle of 30° with the positive x-axis, so its slope is $\tan(30°) = \frac{1}{\sqrt{3}}$. After rotating by 15° clockwise, the new angle is 15°, and the new slope is $\tan(15°) = 2 - \sqrt{3}$. Using the point-slope form at point $A(9, 0)$, we get:$y = (2 - \sqrt{3})(x - 9)$

Expanding and rearranging leads to the equation $\frac{y}{\sqrt{3} - 2} + x = 9$, which matches Option (2).