If z 1- z 2- z 3- z 4 are the four complex numbers represented by the vertices of a quadrilateral taken in order such that z 1- z 2- z 3- z 4 and amp z 4- z 1- z 2- z 1-2- then the quadrilateral is aA- squareB- rhombusC- rectangleD- a cyclic quadrilateral

The correct option is D nbsp;(a cyclic quadrilateral)We have nbsp;z_1 z_4=z_2 z_3⇒ z_1+z_3=z_2+z_4 nbsp;⇒z_1+z_3/2=z_2+z_4/2Thus, the diagonals bisect e ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XI

Mathematics

Terminologies Related to an Ellipse

If z 1, z 2, ...

Question

If $z_{1}, z_{2}, z_{3}, z_{4}$ are the four complex numbers represented by the vertices of a quadrilateral taken in order such that $z_{1} - z_{2} = z_{3} - z_{4}$ and amp $\frac{z_{4} - z_{1}}{z_{2} - z_{1}} = \frac{π}{2}$ , then the quadrilateral is a

square

No worries! We‘ve got your back. Try BYJU‘S free classes today!

rhombus

No worries! We‘ve got your back. Try BYJU‘S free classes today!

rectangle

No worries! We‘ve got your back. Try BYJU‘S free classes today!

a cyclic quadrilateral

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct option is D a cyclic quadrilateral
We have $z_{1} - z_{4} = z_{2} - z_{3}$
$\Rightarrow z_{1} + z_{3} = z_{2} + z_{4}$
$\Rightarrow \frac{z_{1} + z_{3}}{2} = \frac{z_{2} + z_{4}}{2}$
Thus, the diagonals bisect each other.
Here, $z_{1} + z_{3}$ and $z_{2} + z_{4}$ represents the diagonals.
$∴$ It is a parallelogram
Also, $a m p (\frac{z_{4} - z_{1}}{z_{2} - z_{1}}) = \frac{π}{2}$
Here, $z_{4} - z_{1}$ and $z_{2} - z_{1}$ represents sides joined by vertices $z_{4}, z_{1}$ and $z_{2}, z_{1}$ respectively.
thus, angle at $z_{1}$ is a right angle
$∴$ It is a rectangle and hence a cyclic quadrilateral.
Therefore, the correct option is A.

Suggest Corrections

Similar questions

Q. If

z_{1}, z_{2}, z_{3}, z_{4}

are the four complex numbers represented by the vertices of a quadrilateral taken in order such that

z_{1} - z_{4} = z_{2} - z_{3}

and

a m p \frac{z_{4} - z_{1}}{z_{2} - z_{1}} = \frac{π}{2}

then the quadrilateral is a

Q. Let

z_{1}, z_{2}, z_{3}, z_{4}

are distinct complex numbers representing the vertices of a quadrilateral

A B C D

taken in order. If

z_{1} - z_{3} = z_{2} - z_{4}

and

arg (\frac{z_{4} - z_{1}}{z_{2} - z_{1}}) = \frac{π}{2},

then the quadrilateral is

Q. If

z_{1}, z_{2}, z_{3}, z_{4}

are four distinct complex numbers representing the vertices of a quadrilateral taken in order such that

z_{1} - z_{4} = z_{2} - z_{3}

and

a r g (\frac{z_{4} - z_{1}}{z_{2} - z_{1}}) = \frac{π}{2},

, then quadrilateral MAY BE a

z_{1}, z_{2}, z_{3}, z_{4}

are distinct complex numbers representing the vertices of a quadrilateral ABCD taken in order. If

z_{1} - z_{4} = z_{2} - z_{3}

and

a r g [(z_{4} - z_{1}) / (z_{2} - z_{1})] = \frac{π}{2}

, then the quadrilateral is

Q. If

z_{1}, z_{2}, z_{3}, z_{4}

are the four complex numbers repersented by the vertices of a quadrilateral taken in order such that

z_{1} - z_{4} = z_{2} - z_{3}

and

a m p \frac{z_{4} - z_{1}}{z_{2} - z_{1}} = \frac{π}{2}

then the quadrilateral is a

Join BYJU'S Learning Program

If z1,z2,z3,z4 are the four complex numbers represented by the vertices of a quadrilateral taken in order such that z1−z2=z3−z4 and amp z4−z1z2−z1=π2, then the quadrilateral is a

If $z_{1}, z_{2}, z_{3}, z_{4}$ are the four complex numbers represented by the vertices of a quadrilateral taken in order such that $z_{1} - z_{2} = z_{3} - z_{4}$ and amp $\frac{z_{4} - z_{1}}{z_{2} - z_{1}} = \frac{π}{2}$ , then the quadrilateral is a