64.4k views
3 votes
If A and B be two complex numbers satisfying A/B +B/A = 1 Then the two points represented by A and B and the origin form the vertices of

A. An equilateral triangle
B. An isosceles triangle which is not equilateral
C. An isosceles triangle which is not right angled
D. A right angled triangle

1 Answer

7 votes

Final answer:

The complex numbers A and B satisfying the equation A/B + B/A = 1 form an isosceles triangle with the origin as vertices, which is not necessarily equilateral or right angled (Option B).

Step-by-step explanation:

If A and B are two complex numbers satisfying the equation A/B + B/A = 1, we can start by considering what the equation tells us about the geometric relationship between A and B when represented as vectors in the complex plane. The given condition can be rewritten as A2 + B2 = AB.

Squaring both sides of the equation A2 + B2 = AB, we get (A2 + B2)2 = A2B2 which simplifies to A4 + 2A2B2 + B4 = A2B2. Subtracting A2B2 from both sides gives A4 + B4 = A2B2. Subtracting 2A2B2 from both sides gives A4 - 2A2B2 + B4 = 0, which factors into (A2 - B2)2 = 0.

This shows that A2 = B2 and hence |A| = |B|. Since the magnitudes are equal, the points represented by A and B are equidistant from the origin, forming an isosceles triangle. But because A and B are not necessarily equal or opposite, the triangle is not guaranteed to be equilateral or right angled.

Therefore, the correct answer is Option B: An isosceles triangle which is not equilateral.

User BrezzaP
by
8.5k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.