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If x(bc)(b+c2a)=y(ca)(c+a2b)=z(ab)(a+b2c)\frac{x}{(b-c)(b+c-2a)} = \frac{y}{(c-a)(c+a-2b)} = \frac{z}{(a-b)(a+b-2c)} then value of x+y+zx + y + z is:

Solution

Correct Option: 3

Let each ratio equal kk. Then x=k(bc)(b+c2a)x = k(b-c)(b+c-2a), y=k(ca)(c+a2b)y = k(c-a)(c+a-2b), z=k(ab)(a+b2c)z = k(a-b)(a+b-2c). Expanding: x=k(b2c22ab+2ac)x = k(b^2 - c^2 - 2ab + 2ac), y=k(c2a22bc+2ab)y = k(c^2 - a^2 - 2bc + 2ab), z=k(a2b22ac+2bc)z = k(a^2 - b^2 - 2ac + 2bc). Adding, all terms cancel: x+y+z=0x + y + z = 0.

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