11TH•Mathematics•Complex Numbers•Question 14

Solution

For $z=x+iy$, solve the equation $|5z+4+i|=|5z-3+2i|$.

Let $z=x+iy$ .

Given:

|5z+4+i|=|5z-3+2i|

Compute each expression:

5z+4+i=5x+5iy+4+i=(5x+4)+i(5y+1)

5z-3+2i=5x+5iy-3+2i=(5x-3)+i(5y+2)

Square both moduli:

(5x+4)^2+(5y+1)^2=(5x-3)^2+(5y+2)^2

Expand and simplify:

\begin{aligned} (25x^2+40x+16)+(25y^2+10y+1) &= (25x^2-30x+9)+(25y^2+20y+4)\\ 40x+10y+17 &= -30x+20y+13\\ 70x-10y+4 &= 0\\ 35x-5y+2 &= 0 \end{aligned}

So the cartesian form is:

\boxed{35x-5y+2=0}\quad\text{or}\quad\boxed{y=7x+\frac{2}{5}}

forum

menu_bookLesson...

Start discussion

keyboard_arrow_down

Discuss theoretical concepts, mathematical formulas, and architectural foundations.

No comments yet. Be the first to start the discussion!