A line passing through the point A(9, 0) makes an angle of 30° with the positive direction of x-axis. If this line is rotated about A through an angle of 15° in the clockwise direction, then its equations in the new position is [2024]

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Solution

Q.
A line passing through the point A(9, 0) makes an angle of 30° with the positive direction of x-axis. If this line is rotated about A through an angle of 15° in the clockwise direction, then its equations in the new position is [2024]

1
$\frac{x}{\sqrt{3} + 2} + y = 9$

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

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

4
$\frac{y}{\sqrt{3} + 2} + x = 9$

Ans.
(2)

Let $l$ be the line passing through point A(9, 0) making angle 30° with x-axis.

Line is rotated clockwise by 15° then $l'$ is the new position of line where it make 15° angle with x-axis.

So, equation of line passing through (9, 0) and making angle 15° with x-axis is (y – 0) = tan 15° (x – 9)

$\Rightarrow$ y = tan (45° – 30°)(x – 9) = $(\frac{\tan 45 ° - \tan 30 °}{1 + \tan 45 ° \cdot \tan 30 °}) (x - 9)$

$\Rightarrow y = (2 - \sqrt{3}) (x - 9)$

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

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