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Question
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 a and b are equal to
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 a and b are equal to
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Solution
The correct option is A a=b=2−√3
If z1,z2 and z3=0 be the three sides, then the sides are given by |z1−z2|,|z1−0| and |z2−0|
Since the triangle is equilateral the sides are of equal length
∴(a−1)2+(b−1)2=a2+1=b2+1
From last two, we have a=b and from first two, we have
∴a2−4a+1=0∵a=b
∴a=2+√3,2−√3.
Since 0<a<1
∴a=2−√3=b.
Ans: A
If z1,z2 and z3=0 be the three sides, then the sides are given by |z1−z2|,|z1−0| and |z2−0|
Since the triangle is equilateral the sides are of equal length
∴(a−1)2+(b−1)2=a2+1=b2+1
From last two, we have a=b and from first two, we have
∴a2−4a+1=0∵a=b
∴a=2+√3,2−√3.
Since 0<a<1
∴a=2−√3=b.
Ans: A
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