JEE Exam > JEE Questions > If points corresponding to the complex number...
Start Learning for Free
If points corresponding to the complex numbers z1, z2, z3 and z4 are the vertices of a rhombus, taken in order, then for a non-zero real number k
- a)z1 – z3 = i k( z2 –z4)
- b)z1 – z2 = i k( z3 –z4)
- c)z1 + z3 = k( z2 +z4)
- d)z1 + z2 = k( z3 +z4)
Correct answer is option 'A'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
If points corresponding to the complex numbers z1, z2, z3and z4are the...
AC = z3 = z1 eiπ
= z1 (cosπ + i sinπ)
= z3 = z1(-1 + i(0))
= z3 = -z1
AC = z1 - z3
BC = z2 - z4
(z1 - z3)/(z2 - z4) = k
(z1 - z3) = eiπ/2(z2 - z4)
(z1 - z3) k(cosπ/2 + sinπ/2) (z2 - z4)
z1 - z3 = ki(z2 - z4)
z1 - z3 = ik(z2 - z4)
= z1 (cosπ + i sinπ)
= z3 = z1(-1 + i(0))
= z3 = -z1
AC = z1 - z3
BC = z2 - z4
(z1 - z3)/(z2 - z4) = k
(z1 - z3) = eiπ/2(z2 - z4)
(z1 - z3) k(cosπ/2 + sinπ/2) (z2 - z4)
z1 - z3 = ki(z2 - z4)
z1 - z3 = ik(z2 - z4)
Most Upvoted Answer
If points corresponding to the complex numbers z1, z2, z3and z4are the...
We know that the opposite sides of a rhombus are parallel and equal in length. Therefore,
|z2 - z1| = |z4 - z3| (Opposite sides)
Also, the diagonals of a rhombus bisect each other at right angles. Therefore,
(z2 + z1)/2 + (z4 + z3)/2 = 2(z2 + z1 + z4 + z3)/4 = 0 (Diagonals bisect each other)
Simplifying the above equation, we get
z2 + z1 + z4 + z3 = 0
Now, let's consider the given expression k(z1 - z2)(z3 - z4).
Expanding this expression, we get
k(z1z3 - z1z4 - z2z3 + z2z4)
We can express z4 as z3 + (z2 - z1) (since z1z4 is parallel and equal to z2z3)
Substituting this value of z4 in the above expression, we get
k(z1z3 - z1(z3 + z2 - z1) - z2z3 + z2(z3 + z2 - z1))
Simplifying the above expression, we get
k(z1z2 - z1z3 - z2z3 + z2z3) = k(z1z2 - z1z3 - z2z3)
Now, we can express z1 as z2 + (z2 - z3) (since z1z2 is parallel and equal to z3z4)
Substituting this value of z1 in the above expression, we get
k((z2 + (z2 - z3))z2 - (z2 + (z2 - z3))z3 - z2z3) = k(z2^2 - z2z3 + z2^2 - z3z2 - z2z3) = k(2z2^2 - 2z2z3 - z3z2)
Now, we can use the fact that z2 + z1 + z4 + z3 = 0 to simplify the above expression. Substituting z4 as z3 + (z2 - z1) and rearranging the terms, we get
2(z2 - z1)(z3 - z2) = z3z2 - z2^2
Substituting this value in the above expression, we get
k(2z2^2 - 2z2z3 - (z3z2 - z2^2)) = k(z2^2 - 2z2z3 + z3^2)
Now, we can express z4 as z3 + (z1 - z2) (since z1z4 is parallel and equal to z2z3)
Substituting this value of z4 in the above expression, we get
k(z1z3 - z1(z3 + z1 - z2) - z2z3 + z2(z3 + z1 - z2))
Simplifying the above expression, we get
k(z1z2 - z2z3 - z1z2 + z1z3) = k(z1z3 -
|z2 - z1| = |z4 - z3| (Opposite sides)
Also, the diagonals of a rhombus bisect each other at right angles. Therefore,
(z2 + z1)/2 + (z4 + z3)/2 = 2(z2 + z1 + z4 + z3)/4 = 0 (Diagonals bisect each other)
Simplifying the above equation, we get
z2 + z1 + z4 + z3 = 0
Now, let's consider the given expression k(z1 - z2)(z3 - z4).
Expanding this expression, we get
k(z1z3 - z1z4 - z2z3 + z2z4)
We can express z4 as z3 + (z2 - z1) (since z1z4 is parallel and equal to z2z3)
Substituting this value of z4 in the above expression, we get
k(z1z3 - z1(z3 + z2 - z1) - z2z3 + z2(z3 + z2 - z1))
Simplifying the above expression, we get
k(z1z2 - z1z3 - z2z3 + z2z3) = k(z1z2 - z1z3 - z2z3)
Now, we can express z1 as z2 + (z2 - z3) (since z1z2 is parallel and equal to z3z4)
Substituting this value of z1 in the above expression, we get
k((z2 + (z2 - z3))z2 - (z2 + (z2 - z3))z3 - z2z3) = k(z2^2 - z2z3 + z2^2 - z3z2 - z2z3) = k(2z2^2 - 2z2z3 - z3z2)
Now, we can use the fact that z2 + z1 + z4 + z3 = 0 to simplify the above expression. Substituting z4 as z3 + (z2 - z1) and rearranging the terms, we get
2(z2 - z1)(z3 - z2) = z3z2 - z2^2
Substituting this value in the above expression, we get
k(2z2^2 - 2z2z3 - (z3z2 - z2^2)) = k(z2^2 - 2z2z3 + z3^2)
Now, we can express z4 as z3 + (z1 - z2) (since z1z4 is parallel and equal to z2z3)
Substituting this value of z4 in the above expression, we get
k(z1z3 - z1(z3 + z1 - z2) - z2z3 + z2(z3 + z1 - z2))
Simplifying the above expression, we get
k(z1z2 - z2z3 - z1z2 + z1z3) = k(z1z3 -
Free Test
FREE
| Start Free Test |
Community Answer
If points corresponding to the complex numbers z1, z2, z3and z4are the...
Option a is correct answer
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer?
Question Description
If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer?.
If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer?.
Solutions for If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer?, a detailed solution for If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If points corresponding to the complex numbers z1, z2, z3and z4are the vertices of a rhombus, taken in order, then for a non-zero real number ka)z1– z3= i k( z2–z4)b)z1– z2= i k( z3–z4)c)z1+ z3= k( z2+z4)d)z1+ z2= k( z3+z4)Correct answer is option 'A'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup