Question

If the line joining two points $A(2,0)$ and $B(3,1)$ is rotated about $A$ in anticlockwise direction through an angle of $15^\circ$, then the equation of the line in its new position is

• A. $y = \sqrt{3}x - 6$
• B. $y = \sqrt{3}x - 2\sqrt{3}$
• C. $y = -\sqrt{3}x + 2\sqrt{3}$
• D. $y = \frac{1}{\sqrt{3}}x - 2\sqrt{3}$

Hint:

Remember that a slope of 1 always corresponds to an angle of $45^\circ$. Adding $15^\circ$ brings it to $60^\circ$, which has a well-known slope of $\sqrt{3}$. Since the line must vanish at $x=2$ (because $y=0$ at point $A$), only option (B) satisfies $\sqrt{3}(2) - 2\sqrt{3} = 0$!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
A straight line segment initially passes through two given points $A(2,0)$ and $B(3,1)$. This line is then rotated about the pivot point $A$ in an anticlockwise direction by an angle of $15^\circ$. We need to find the equation of the line in this new transformed configuration.

Step 2: Key Formula or Approach:
1. Find the initial inclination angle $\theta_1$ of line $AB$ using its slope formula: $$m_1 = \tan \theta_1 = \frac{y_2 - y_1}{x_2 - x_1}$$ 2. The new inclination angle after an anticlockwise rotation of $\alpha = 15^\circ$ is $\theta_2 = \theta_1 + 15^\circ$. 3. Use the point-slope form equation with the fixed pivot point $A(2,0)$ to get the final line equation: $$y - y_1 = m_2(x - x_1)$$

Step 3: Detailed Explanation:
First, let's calculate the slope of the original line passing through $A(2,0)$ and $B(3,1)$: $$m_1 = \frac{1 - 0}{3 - 2} = \frac{1}{1} = 1$$ Since $\tan \theta_1 = 1$, the initial inclination angle is: $$\theta_1 = 45^\circ$$ The line is rotated anticlockwise by $15^\circ$, which increases the angle of inclination: $$\theta_2 = 45^\circ + 15^\circ = 60^\circ$$ The slope of the line in its new position is: $$m_2 = \tan 60^\circ = \sqrt{3}$$ Since the line is rotated about point $A$, it still passes through $A(2,0)$. Using the point-slope form: $$y - 0 = \sqrt{3}(x - 2)$$ $$y = \sqrt{3}x - 2\sqrt{3}$$ Rearranging into standard equation form gives: $$\sqrt{3}x - y - 2\sqrt{3} = 0$$ This matches perfectly with the expression presented in option (B).

Step 4: Final Answer:
The equation of the line in its new position is $y = \sqrt{3}x - 2\sqrt{3}$, which corresponds to option (B).