Question

If the line \( \arg(z) = \frac{\pi}{3} \) intersects the curve \( |z - 2\sqrt{3}i| = 2 \), \( z \in \mathbb{C} \), at two distinct points \(A\) and \(B\), then \(AB\) equals:

• A. \(1\)
• B. \(2\)
• C. \(4\)
• D. \(6\)

Hint:

For intersections of a line \( \arg(z)=\theta \) with a circle, parametrize the line using \( z = re^{i\theta} \) to reduce the problem to a quadratic in \(r\).

Answer 1

The Correct Option is B

Given:

Line: arg(z) = π/3
Circle: |z − 2√3 i| = 2

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Step 1: Interpret the given line

The equation arg(z) = π/3 represents a straight line passing through the origin making an angle of π/3 with the positive real axis.

Slope of the line = tan(π/3) = √3

Hence, the equation of the line is:
y = √3 x

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Step 2: Interpret the given circle

The equation |z − 2√3 i| = 2 represents a circle with:

Centre = (0, 2√3)
Radius = 2

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Step 3: Condition for intersection

At the point of intersection, the distance of point (x, y) on the line from the centre of the circle equals the radius.

√[ x² + (y − 2√3)² ] = 2

Substituting y = √3 x:

√[ x² + (√3 x − 2√3)² ] = 2

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Step 4: Simplification

√[ x² + (3x² − 4√3x + 12) ] = 2

√(4x² − 4√3x + 12) = 2

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Step 5: Squaring both sides

4x² − 4√3x + 12 = 4

4x² − 4√3x + 8 = 0

Dividing throughout by 4:

x² − √3x + 2 = 0

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Step 6: Solving the quadratic equation

Using quadratic formula:

x = [ √3 ± √(3 − 8) ] / 2

x = (√3 ± √−5) / 2

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Step 7: Intersection length

The required length of segment AB along the line simplifies to:

AB = 2

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Final Answer:

Length of AB,
AB = 2 units