Question

If the points representing the complex numbers $-4+3i,2-3i,0+pi$ are collinear, then the value of $p$ is

• A. $1$
• B. $-1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $1$

Given that,

$z_1=-4+3i$

$z_2=2-3i$

$z_3=0+pi$

we know that

$z=x+iy$

so,

points $(x_1,y_1)=(-4,3)$

$(x_2,y_2)=(2,-3)$

$(x_3,y_3)=(0,p)$

So,

The points are collinear

then,

Area of $\Delta =0$

$\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 0\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{ccc}-4& 3& 1\\ 2& -3& 1\\ 0& p& 1\end{array}\right|=0$

$\Rightarrow -4(-3x)-p\times 1)-3(2\times 1-1\times 0)+(2\times p-(-3)\times 0)=0$

$\Rightarrow 12+4p-6+2p=0$

$\Rightarrow 6p+6=0$

$\Rightarrow 6p=-6$

$p=-1$

Hence, this is the anwer.