JAMIA 2026 — Mathematics PYQ

Method 1: Algebraic Method

Given:

$$z=x+iy$$

The given equation representing the locus is:

$$|z+1|=|z+i|$$

Substitute the value of $z$ into the equation:

$$|(x+iy)+1|=|(x+iy)+i|$$

Group the real parts and imaginary parts together on both sides:

$$|(x+1)+iy|=|x+i(y+1)|$$

The modulus of a complex number $a+ib$ is defined as $\sqrt{a^2+b^2}$. Applying this definition to both sides gives:

$$\sqrt{(x+1{)}^2+y^2}=\sqrt{x^2+(y+1{)}^2}$$

Square both sides of the equation to eliminate the radical square roots:

$$(x+1{)}^2+y^2=x^2+(y+1{)}^2$$

Expand both quadratic expressions using the algebraic identity $(A+B{)}^2=A^2+2AB+B^2$:

$$(x^2+2x+1)+y^2=x^2+(y^2+2y+1)$$

Simplify the equation by canceling out the common terms ($x^2$, $y^2$, and $1$) present on both sides:

$$2x=2y$$

Divide both sides by $2$:

$$x=y$$

$$x-y=0$$

This linear equation represents a straight line passing through the origin.