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

The Correct Option is B

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).