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If a and b are real numbers between 0 and 1 such that points (a,1), (1,b) and (0,0) form an equilateral triangle then find the values of a and b respectively.

If a and b are real numbers between 0 and 1 such that points (a,1), (1,b) and (0,0) form an equilateral triangle then find the values of a and b  respectively.

Grade:11

1 Answers

Arun
25763 Points
2 years ago
If we think of this question from complex number perspective
 
z1 = a + i
z2 = 1 + bi
z3 = 0
 
Now
 
As z1, z2, z3 form an equilateral triangle :
z1^2 + z2^2 + z3^2 = z1.z2 + z2.z3 + z3.z1
a^2 - 1 + 2ai + 1 - b^2 + 2bi = a + abi + i - b
Hence, a^2 - b^2 = a - b & 2a + 2b = ab + 1
Hence, a = b from first equation..
So, 2a + 2a = a^2 + 1
a^2 - 4a + 1 = 0
Solve this equation to get a and b.
 
 
Regards
Arun (askIITians forum expert)

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