Question

IfX-y=c,z-x=b,y-z=a and a^2+b^2+c^2=25 then x^2+y^2+z^2-xy-yz-zx=

Answer 1

Step-by-step explanation:

apply the values of a,b nd c in a^2 + b^2 + c^2 = 25

Answer 2

Answer:

$x {}^{2} + y {}^{2} + z {}^{2} - xz - yz - zx = 12.5$

Answer is 12.5

Step-by-step explanation:

Given,

$x - y = c \\ z - x = b \\ y - z = a$

Adding all the three equations we get :

$x - y + z - x + y - z = a + b + c \\ 0 = a + b + c$

Now,

$(a + b + c) {}^{2} = a {}^{2} + b {}^{2} + c {}^{2} + 2ab + 2bc + 2ca \\ (0) {}^{2} = 25 + 2(ab + bc + ca) \\ 0 - 25 = 2(ab + bc + ca) \\ - 25 \div 2 = ab + bc + ca$

Now substituting Values of a, b, c in above equation we get,

$- 12.5 = (y - z)(z - x) + (z - x)(x - y) + (x - y)(y - z) \\ - 12.5 = yz - xy - z {}^{2} + xz - zy - x {}^{2} + xy + xy - xz - y {}^{2} + yz \\ - 12.5 = - x {}^{2} - y {}^{2} - z {}^{2} + xy + yz + zx \\ - 12.5 = - (x {}^{2} + y {}^{2} + z {}^{2} - xy - yz - zx) \\ 12.5 = x {}^{2} + y {}^{2} + z {}^{2} - xy - yz - zx$

Hope You Got the answer. :)