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Let x, y, z be positive real number such that x^2+y^2=49, y^2+y z+z^2=36 and x^2+ 3 x z+z^2=25 . If the value of 2 x y+ 3 y z+z x can be written as p q where p, q are integers and q is not divisible by square of any prime number, find p + q .

Correct answer

30

Step-by-step solution

aligned & x^2+y^2=49 & x^2+y^2-49 2 x y =0 & A=0, A=90^ & in B= y^2+z^2-36 2 y z =- 1 2 , B=120^ & C= z^2+x^2-25 2 x z =- 3 2 , C=150^ aligned aligned & Area of PQR = 9 4 3 2 =6 6 & 2 xy + 3 yz + zx =4 Area of PQR & =4 6 6 =24 6 & 24+6=30 aligned

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