Question

Let \( z=x+iy \) and \( P(x,y) \) be a point on the Argand plane. If \( z \) satisfies the condition \( \text{Arg}\left(\frac{z-3i}{z+2i}\right) = \frac{\pi}{4} \), then the locus of P is

• A. \( x^2+y^2-y-6=0, (x,y) \neq (0,-2) \)
• B. \( x^2+y^2-x-y-6=0, (x,y) \neq (0,-2) \)
• C. \( x^2+y^2+5x-y-6=0, (x,y) \neq (0,-2) \)
• D. \( x^2+y^2+x-y-6=0, (x,y) \neq (0,-2) \)

Hint:

For locus problems involving \( \text{Arg}((z-z_1)/(z-z_2)) = \theta \), the curve is a circle passing through \( z_1 \) and \( z_2 \). You can often eliminate options by simply checking if the coordinates of \( z_1 \) and \( z_2 \) satisfy the given equations.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:

The equation Arg((z - z1) / (z - z2)) = α represents an arc of a circle passing through z1 and z2. Here z1 = 3i and z2 = -2i.

Step 2: Key Formula or Approach:

The points A(0,3) and B(0,-2) lie on the locus. We can check which option satisfies these points.

Substitute (0,3):
0 + 9 + 0 - 3 - 6 = 0   (Satisfied)

Substitute (0,-2):
0 + 4 + 0 - (-2) - 6 = 0   (Satisfied)

All options might satisfy this, so we need the full equation.
Since the angle is π/4, the center (h,k) forms a right angle with the chord AB at the center.

Step 3: Detailed Explanation:

The chord length AB = |3i - (-2i)| = 5.

If R is the radius, then:
R² + R² = AB²
2R² = 25
R² = 12.5

The perpendicular bisector of AB is:
y = (3 + (-2)) / 2 = 0.5
So, k = 0.5.

The distance from center to A is R:
h² + (3 - 0.5)² = 12.5
h² + 6.25 = 12.5
h² = 6.25
h = ±2.5

So, the equation is:
(x - h)² + (y - 0.5)² = 12.5

Expanding:
x² - 2hx + h² + y² - y + 0.25 = 12.5
x² + y² - 2hx - y + 6.5 = 12.5
x² + y² - 2hx - y - 6 = 0

Now we determine the sign of -2h.
For Arg = π/4 > 0, the locus is on one side of the chord. Using the standard orientation or checking a point, the coefficient of x is +5.

Thus, the equation is:
x² + y² + 5x - y - 6 = 0

Step 4: Final Answer:

The locus is x² + y² + 5x - y - 6 = 0.