JEE 2023 — Mathematics PYQ

To solve this, we will convert the complex equation into the standard Cartesian form of a circle.

Step 1: Simplify the given equation

The given equation is:

Multiply both sides by $|z-3|$:

Step 2: Convert to Cartesian coordinates

Let $z=x+iy$. Substituting this into the equation:

Square both sides to remove the absolute value (modulus) radicals:

Step 3: Expand and rearrange

Expand the squares:

Move all terms to one side:

Divide the entire equation by 3 to get the standard form:

Step 4: Find the Center $(\alpha ,\beta )$ and Radius $\gamma$

For a circle equation $x^2+y^2+2gx+2fy+c=0$:

• Center is $(-g,-f)$

• Radius is $\sqrt{g^2+f^2-c}$

Comparing our equation:

• $2g=-\frac{20}{3}⟹g=-\frac{10}{3}$

• $2f=0⟹f=0$

• $c=\frac{32}{3}$

Center $(\alpha ,\beta )$:

Radius $\gamma$:

Step 5: Calculate the final value

We need to find $3(\alpha +\beta +\gamma )$:

Correct Option: (2)