Question

If \( P(x, y) \) represents the complex number \( z = x + iy \) in the Argand plane and \[ \arg \left( \frac{z - 3i}{z + 4} \right) = \frac{\pi}{2}, \] then the equation of the locus of \( P \) is:

• A. \( x^2 + y^2 + 4x - 3y = 0 \) and \( 3x - 4y>0 \)
• B. \( x^2 + y^2 + 4x - 3y = 0 \) and \( 3x - 4y>0 \)
• C. \( x^2 + y^2 + 4x - 3y + 2 = 0 \) and \( 3x - 4y<0 \)
• D. \( x^2 + y^2 + 4x - 3y + 2 = 0 \) and \( 3x - 4y<0 \)

Hint:

For argument-based locus problems, express the given condition in terms of \( x \) and \( y \) and simplify algebraically to obtain the required equation.

Answer 1

The Correct Option is C

Step 1: Understanding the given equation 
We are given the argument condition: \[ \arg \left( \frac{z - 3i}{z + 4} \right) = \frac{\pi}{2}. \] This implies that the complex number \( \frac{z - 3i}{z + 4} \) is purely imaginary. 

Step 2: Expressing in Cartesian form 
Let \( z = x + iy \), then: \[ \frac{z - 3i}{z + 4} = \frac{(x + iy) - (0 + 3i)}{(x + iy) + (-4 + 0i)} = \frac{x + i(y - 3)}{x - 4 + iy}. \] Multiplying numerator and denominator by the conjugate of the denominator: \[ \frac{(x + i(y - 3))(x - 4 - iy)}{(x - 4 + iy)(x - 4 - iy)}. \] The denominator simplifies to: \[ (x - 4)^2 + y^2. \] The numerator expands as: \[ (x^2 - 4x + ixy - i4y) + i(y - 3)x - i(y - 3)i y. \] Setting the real part to zero gives the equation of the locus. 

Step 3: Finding the locus equation 
After simplification, the equation of the locus is: \[ x^2 + y^2 + 4x - 3y + 2 = 0. \] The given argument condition also implies a restriction on the region of the plane: \[ 3x - 4y<0. \] 

Step 4: Conclusion 
Thus, the correct equation of the locus is: \[ x^2 + y^2 + 4x - 3y + 2 = 0 \quad \text{and} \quad 3x - 4y<0. \]