Question

If x, y and z are distinct real numbers such that x:(y+z)=y:(z+x), then
A) x, y, z are all positive B) xy+yz+zx+1=0
C) x+y+z=0 D) x, y, z are all negative

Answer 1

x/(y+z) = y/( z+x)

Apply C and D

x+ y + z)/ x-y-z = y+ z+ x)/ y- z-x

Now here x+ y + z= 0

Or

x- y - z = y - z - x

2x - 2y = 0

x= y

As x,y,z are distinct so eliminate

Now

If x+ y + z= 0

( x+ y+z)^2 = x^2 + y^2 + z^2 +2(xy+yz+zx)

As minimum value of x^2 + y^2 + z^2 = 0

and So 2( xy + yz+ zx) = 0

xy + yz + zx = 0( minimum value)

which can't be -1

So Only A) correct