Question

Let circle $C$ be the image of
$$ x^2 + y^2 - 2x + 4y - 4 = 0 $$
in the line
$$ 2x - 3y + 5 = 0 $$
and $A$ be the point on $C$ such that $OA$ is parallel to the x-axis and $A$ lies on the right-hand side of the centre $O$ of $C$.
If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $\frac{1}{6}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to: 

• A. 3
• B. \( 3 + \sqrt{3} \)
• C. \( 4 - \sqrt{3} \)
• D. 4

Hint:

To find the distance between two points on a circle, use the arc length formula. The angle subtended by the arc at the center is related to the arc length and the radius. Use the parametric equations of the circle to find the coordinates of points on the circle.

Answer 1

The Correct Option is D

Step 1: Reflecting the center of the circle 

The center of the circle is \( (1, -2) \), and we reflect it across the line \( 2x - 3y + 5 = 0 \). Using the reflection formula, we get: \[ x' = -2, \quad y' = 4. \] Therefore, the new center of the circle after reflection is \( (-3, 4) \).

Step 2: Equation of the reflected circle

The equation of the reflected circle is: \[ (x + 3)^2 + (y - 4)^2 = 9. \]

Step 3: Finding the area of the sector

The angle \( \theta \) subtended by the arc is: \[ \theta = \frac{1}{6} \times 2\pi = \frac{\pi}{3}. \]

Step 4: Length of the chord \( AB \)

The length of the chord \( AB \) is: \[ AB = 2r \sin\left(\frac{\theta}{2}\right) = 6 \times \frac{1}{2} = 3. \]

Final Answer:

The correct option is \( \boxed{4} \).