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If a and b are real numbers between 0 and 1 such that the points z1=a + i , z2= 1 + bi and z3=0 form an equilateral triangle, then find the value of a and b.
As z1, z2, z3 form an equilateral triangle :z1^2 + z2^2 + z3^2 = z1.z2 + z2.z3 + z3.z1a^2 - 1 + 2ai + 1 - b^2 + 2bi = a + abi + i - bHence, a^2 - b^2 = a - b & 2a + 2b = ab + 1Hence, a = b from first equation..So, 2a + 2a = a^2 + 1a^2 - 4a + 1 = 0Solve this equation to get a and b.ThanksBharat BajajIIT Delhiaskiitians faculty
|a+i| = |1+bi| means a^2+1 = 1+b^2 or a=b. This means z1 and z2 are symmetrically placed on either side of the line x=y. Since angle between z1 and z2 is 60 degrees, this means arg (1+bi) = 15 degrees. so that b = a = tan 15 degrees = 2 - sqrt(3)
|a+i| = |1+bi| means a^2+1 =1+b^2 or a^2=b^2 so a=b (since a,b>0) This means (a,1) and (1,b) are symmetric about the line x=y and since the angle between z1 and x2 is 60 degrees, that means arg z2 = arg (1+ib) = 15 degrees so that b = a = tan 15 = 2-sqrt(3)
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