Question

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

Answer 1

distance between (a,1) and (0,0)

√ a^2 + 1

distance between (0,0) (1,b)

√b^2 + 1

distance between(a,1)( 1,b)

√ ( a-1)^2 + ( 1 - b)^2

As it forms an equilateral

so distance between any 2 points are equal

√a^2 +1 = √ b^2 +1 = √ ( a-1)^2 + ( 1 - b)^2

Squaring

a^2 + 1 = b^2 +1 = ( a-1)^2 + ( 1 - b)^2

a = +-b

a^2 = ( a-1)^2 + ( 1 - b)^2

take b=a

a^2 = 2( a-1)^2

a^2 = 2 a^2 + 2 - 4a

a^2 - 4a + 2 = 0

a = 4 +- √ 16 -4)/ 2 = 4 +-√12)/2

= 4+- 2√3)/2

= 2 +-√3

b= 2 +-√3