Question

If \( z = x + iy \) and if the point \( P \) in the Argand diagram represents \( z \), then the locus of the point \( P \) satisfying the equation \( 2|z - 2 - 3i| = 3|z + i - 2| \) is a circle with centre:

• A. \( (10, -21) \)
• B. \( (-10, 21) \)
• C. \( \left(2, -\frac{21}{5}\right) \)
• D. \( \left(-2, \frac{21}{5}\right) \)

Hint:

To find the locus defined by modulus equations in complex numbers, convert the complex number into coordinates \( z = x + iy \), simplify using distance formula, and complete the square to find the circle’s equation.

Answer 1

The Correct Option is C

Step 1: Let \( z = x + iy \). We are given the equation: \[ 2|z - (2 + 3i)| = 3|z - (2 - i)|. \] Step 2: Substitute \( z = x + iy \) into the equation: \[ 2|x + iy - 2 - 3i| = 3|x + iy - 2 + i|. \] Step 3: Simplify both sides: \[ 2\sqrt{(x - 2)^2 + (y - 3)^2} = 3\sqrt{(x - 2)^2 + (y + 1)^2}. \] Step 4: Square both sides to eliminate square roots: \[ 4[(x - 2)^2 + (y - 3)^2] = 9[(x - 2)^2 + (y + 1)^2]. \] Step 5: Let \( A = (x - 2)^2 \). Expand and simplify: \[ 4[A + (y - 3)^2] = 9[A + (y + 1)^2]. \] Now expand both sides: \[ 4[A + y^2 - 6y + 9] = 9[A + y^2 + 2y + 1]. \] \[ 4A + 4y^2 - 24y + 36 = 9A + 9y^2 + 18y + 9. \] Step 6: Bring all terms to one side: \[ (4A - 9A) + (4y^2 - 9y^2) - 24y - 18y + (36 - 9) = 0. \] \[ -5A - 5y^2 - 42y + 27 = 0. \] Step 7: Divide the entire equation by \( -5 \): \[ A + y^2 + \frac{42}{5}y - \frac{27}{5} = 0. \] Recall \( A = (x - 2)^2 \), so: \[ (x - 2)^2 + y^2 + \frac{42}{5}y = \frac{27}{5}. \] Step 8: Complete the square in \( y \): \[ (x - 2)^2 + \left(y + \frac{21}{5}\right)^2 = \frac{27}{5} + \left(\frac{21}{5}\right)^2. \] \[ = \frac{27}{5} + \frac{441}{25} = \frac{135 + 441}{25} = \frac{576}{25}. \] So, the equation of the circle is: \[ (x - 2)^2 + \left(y + \frac{21}{5}\right)^2 = \left(\frac{24}{5}\right)^2. \] Hence, the centre is \( \left(2, -\frac{21}{5}\right) \).