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Question

If (0,0),(a,2) and (2,b) are the vertices of an equilateral triangle such that both a and b lie in (0,2), then the value of 4(a+b)ab is

A
4
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B
4
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C
23
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D
43
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Solution

The correct option is A 4
Let the given points be A(0,0),B(a,2),C(2,b)
Since, A,B,C are the vertices of equilateral triangle,
AB=BC=AC

By using distance formula, we have
AB=a2+4 (1)

Similarly, BC=(a2)2+(b2)2 (2)
and AC=4+b2 (3)

Equating equation (1) and (3), we get
4+a2=4+b2
a2+4=b2+4
a=±b
Since a and b lie in (0,2),
Both a and b are positive.
a=b

Now, 4(a+b)ab=8aa2

Equating equation (1) and (2), we get
(a2)2+(b2)2=a2+4a28a+4=0

Hence, 4(a+b)ab=8aa2=4

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