Question

If $P,Q,R,S$ are represented by the complex numbers $4+I,1+6i,-4+3i,-1-2i$ respectively, then $PQRS$ is a

• A. Rectangle
• B. Square
• C. Rhombus
• D. Parallelogram

Answer 1

The correct option is B Square

$P\to 4+i$

$Q\to 1+6i$

$R\to 1+6i$

$S\to -1-2i$

We represent these points in arg and plane.

REF.Image

$PQ=\sqrt{(4-1{)}^2+(1-6{)}^2}=\sqrt{34}$

$QR=\sqrt{(1+4{)}^2+(6-3{)}^2}=\sqrt{34}$

$RS=\sqrt{(-+41{)}^2+(-2-3{)}^2}=\sqrt{34}$

$SP=\sqrt{(4+1{)}^2+(1+2{)}^2}=\sqrt{34}$

All sides are equal,

It must be square as rhombus

Now measure angle between PQ & QR

If it is ${90}^{\circ }$ then it is square otherwise rhombus slope of $PQ=\frac{6-1}{1-4}=\frac{5}{-3}=\frac{-5}{3}=m_1$

slope of $QR=\frac{3-6}{-4-1}=\frac{-3}{-5}=\frac{3}{5}=m_2$

$m_1{m}_2=-1$

Angle is ${90}^{\circ }$

$9+$ is square

B is correct