Question

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

Answer 1

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.