aligned & If (x₁-x₂ )^2+ (y₁-y₂ )^2=a^2 & (x₂-x₃ )^2+ (y₂-y₃ )^2=b^2 & and (x₃-x₁ )^2+ (y₃-y₁ )^2=c^2 aligned then | array lll x₁ & y₁ & 1 x₂ & y₂ & 1 x₃ & y₃ & 1 array |^2=(a+b+c)(b+c-a)(c+a-b)(a+b-c) . Find the value of .

Step-by-step solution

First 3 equations are suggestive that ( x ₁, y ₁ ) ( x ₂, y ₂ ) and ( x ₃, y ₃ ) and the vertices of a BCA Now .A= 1 2 | array lll x ₁ & y ₁ & 1 x ₂ & y ₂ & 1 x ₃ & y ₃ & 1 array | , 4 ~A ^2= | array lll x ₁ & y ₁ & 1 x ₂ & y ₂ & 1 x ₃ & y ₃ & 1 array |^2 aligned & 4 A^2=4 s(s-a)(s-b)(s-c) & 16 A^2=2 s(2 s-2 a)(2 s-2 b)(2 s-2 c)=R H S aligned aligned & and LHS =(16) ( 1 4 ) | array lll x₁ & y₁ & 1 x₂ & y₂ & 1 x₃ & y₃ & 1 array |^2 & 4 | array lll x₁ & y₁ & 1 x₂ & y₂ & 1 x₃ & y₃ & 1 array |^2 aligned =4 Ans.]