## Tag Archives: KVPY

### Geometry with complex numbers — section formula

It ain’t complex, it’s simple !!

Section Formula:

If $P(z)$ divides the line segment joining $A(z_{1})$ and $B(z_{2})$ internally in the ratio $m:n$, then

$z = \frac{mz_{2}+nz_{1}}{m+n}$

If the division is external, then $z=\frac{mz_{2}-nz_{1}}{m-n}$

Proof:

Let $z_{1}=x_{1}+iy_{1}$, $z_{2}=x_{2}+iy_{2}$. Then, $A \equiv (x_{1},y_{1})$ and $B \equiv (x_{2},y_{2})$.

Let $z = x+iy$. Then, $P \equiv (x,y)$. We know from co-ordinate geometry,

$x = \frac{mx_{2}+nx_{1}}{m+n}$ and $y=\frac{my_{2}+my_{1}}{m+n}$

Hence, complex number of P is

$z = \frac{mx_{2}+nx_{1}}{m+n}+i\frac{my_{2}+my_{1}}{m+n}$

$\frac{m(x_{2}+iy_{2})+n(x_{1}+iy_{1})}{m+n}$

$mz_{2}+nz_{1}$

more later,

Nalin Pithwa