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Let z₁ and z₂ be two roots of the equation z2 + az + b = 0,z being complex . Further ,assume that the origin ,z₁ and z₂ form an equilateral triangle .Then
- a)a2= b
- b)a2 = 2b
- c)a2 = 3b
- d)a2 = 4b
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Let z and z be two roots of the equation z2 + az + b = 0,z being compl...
Solution:
Let's assume that the roots of the given equation are z1 and z2. We also know that the origin, z1, and z2 forms an equilateral triangle.
Using the distance formula, we can find the distance between z1 and z2 as follows:
|z1 - z2| = √[(Re(z1) - Re(z2))^2 + (Im(z1) - Im(z2))^2]
Since the triangle is equilateral, the distance between z1 and z2 is equal to the distance between the origin and z1 (or z2). Therefore,
|z1 - z2| = |z1| = |z2|
Squaring both sides, we get:
|z1 - z2|^2 = |z1|^2
Expanding the left side using the distance formula, we get:
[(Re(z1) - Re(z2))^2 + (Im(z1) - Im(z2))^2] = |z1|^2
Since z1 and z2 are roots of the equation z2 - az + b = 0, we have:
z1^2 - az1 + b = 0
z2^2 - az2 + b = 0
Adding these equations, we get:
(z1^2 + z2^2) - a(z1 + z2) + 2b = 0
Using the identity (z1 + z2)^2 = z1^2 + z2^2 + 2z1z2, we can rewrite the above equation as:
(z1 + z2)^2 - 3z1z2 - a(z1 + z2) + 2b = 0
Since z1 and z2 are the roots of the equation, we have:
z1z2 = b
Substituting this in the above equation, we get:
(z1 + z2)^2 - 3b - a(z1 + z2) + 2b = 0
(z1 + z2)^2 - a(z1 + z2) - b = 0
(z1 + z2)^2 = a(z1 + z2) + b
Using the identity (z1 + z2) = -a, we can rewrite the above equation as:
(-a)^2 = a(-a) + b
a^2 = b
Therefore, the correct answer is option C, a^2 = 3b.
Let's assume that the roots of the given equation are z1 and z2. We also know that the origin, z1, and z2 forms an equilateral triangle.
Using the distance formula, we can find the distance between z1 and z2 as follows:
|z1 - z2| = √[(Re(z1) - Re(z2))^2 + (Im(z1) - Im(z2))^2]
Since the triangle is equilateral, the distance between z1 and z2 is equal to the distance between the origin and z1 (or z2). Therefore,
|z1 - z2| = |z1| = |z2|
Squaring both sides, we get:
|z1 - z2|^2 = |z1|^2
Expanding the left side using the distance formula, we get:
[(Re(z1) - Re(z2))^2 + (Im(z1) - Im(z2))^2] = |z1|^2
Since z1 and z2 are roots of the equation z2 - az + b = 0, we have:
z1^2 - az1 + b = 0
z2^2 - az2 + b = 0
Adding these equations, we get:
(z1^2 + z2^2) - a(z1 + z2) + 2b = 0
Using the identity (z1 + z2)^2 = z1^2 + z2^2 + 2z1z2, we can rewrite the above equation as:
(z1 + z2)^2 - 3z1z2 - a(z1 + z2) + 2b = 0
Since z1 and z2 are the roots of the equation, we have:
z1z2 = b
Substituting this in the above equation, we get:
(z1 + z2)^2 - 3b - a(z1 + z2) + 2b = 0
(z1 + z2)^2 - a(z1 + z2) - b = 0
(z1 + z2)^2 = a(z1 + z2) + b
Using the identity (z1 + z2) = -a, we can rewrite the above equation as:
(-a)^2 = a(-a) + b
a^2 = b
Therefore, the correct answer is option C, a^2 = 3b.
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Let z and z be two roots of the equation z2 + az + b = 0,z being complex . Further ,assume that the origin ,z and z form an equilateral triangle .Thena)a2= bb)a2 = 2bc)a2 = 3bd)a2 = 4bCorrect answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let z and z be two roots of the equation z2 + az + b = 0,z being complex . Further ,assume that the origin ,z and z form an equilateral triangle .Thena)a2= bb)a2 = 2bc)a2 = 3bd)a2 = 4bCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let z and z be two roots of the equation z2 + az + b = 0,z being complex . Further ,assume that the origin ,z and z form an equilateral triangle .Thena)a2= bb)a2 = 2bc)a2 = 3bd)a2 = 4bCorrect answer is option 'C'. Can you explain this answer?.
Let z and z be two roots of the equation z2 + az + b = 0,z being complex . Further ,assume that the origin ,z and z form an equilateral triangle .Thena)a2= bb)a2 = 2bc)a2 = 3bd)a2 = 4bCorrect answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let z and z be two roots of the equation z2 + az + b = 0,z being complex . Further ,assume that the origin ,z and z form an equilateral triangle .Thena)a2= bb)a2 = 2bc)a2 = 3bd)a2 = 4bCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let z and z be two roots of the equation z2 + az + b = 0,z being complex . Further ,assume that the origin ,z and z form an equilateral triangle .Thena)a2= bb)a2 = 2bc)a2 = 3bd)a2 = 4bCorrect answer is option 'C'. Can you explain this answer?.
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Let z and z be two roots of the equation z2 + az + b = 0,z being complex . Further ,assume that the origin ,z and z form an equilateral triangle .Thena)a2= bb)a2 = 2bc)a2 = 3bd)a2 = 4bCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for Let z and z be two roots of the equation z2 + az + b = 0,z being complex . Further ,assume that the origin ,z and z form an equilateral triangle .Thena)a2= bb)a2 = 2bc)a2 = 3bd)a2 = 4bCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of Let z and z be two roots of the equation z2 + az + b = 0,z being complex . Further ,assume that the origin ,z and z form an equilateral triangle .Thena)a2= bb)a2 = 2bc)a2 = 3bd)a2 = 4bCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
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