If `|(x^k,x^(k+2),x^(k+3)), (y^k,y^(k+2),y^(k+3)), (z^k,z^(k+2),z^(k+3))|=(x-y)(y-z)(z-x){1/x+1/y+1/z}` then `k=`

Select the correct options from the given alternatives.If |(x^k,x^(k+2),x^(k+3)),(y^k,y^(k+2),y^(k+3)),(z^k,z^(k+2),z^(k+3))|=(x-y)(y-z)(z-x)(1/x+1/y+1/z) then

Let x gt, 0, y gt 0, z gt 0 are respectively the 2^(nd), 3^(rd), 4^(th) terms of a G.P. and Delta = |(x^(k),x^(k+1),x^(k+2)),(y^(k),y^(k+1),y^(k+2)),(z^(k),z^(k+1),z^(k+2))|=(r-1)^(2)(1-(1)/(r^(2))) (where r is the common ratio), then

det [[x ^ (k), x ^ (k + 2), x ^ (k + 3) y ^ (k), y ^ (k + 2), y ^ (k + 3) z ^ (k) , z ^ (k + 2), z ^ (k + 3)]] = (xy) (yz) (zx) {(1) / (x) + (1) / (y) + (1) / ( from)}

Let x gt 0 , y gt 0 , z gt 0 are respectively the 2^(nd) , 3^(rd) , 4^(th) terms of a G.P. and Delta=|{:(x^(k),x^(k+1),x^(k+2)),(y^(k),y^(k+1),y^(k+2)),(z^(k),z^(k+1),z^(k+2)):}|=(r-1)^(2)(1-(1)/(r^(2))) (where r is the common ratio), then

Let x gt 0 , y gt 0 , z gt 0 are respectively the 2^(nd) , 3^(rd) , 4^(th) terms of a G.P. and Delta=|{:(x^(k),x^(k+1),x^(k+2)),(y^(k),y^(k+1),y^(k+2)),(z^(k),z^(k+1),z^(k+2)):}|=(r-1)^(2)(1-(1)/(r^(2))) (where r is the common ratio), then

Let x gt 0 , y gt 0 , z gt 0 are respectively the 2^(nd) , 3^(rd) , 4^(th) terms of a G.P. and Delta=|{:(x^(k),x^(k+1),x^(k+2)),(y^(k),y^(k+1),y^(k+2)),(z^(k),z^(k+1),z^(k+2)):}|=(r-1)^(2)(1-(1)/(r^(2))) (where r is the common ratio), then

Let x gt 0 , y gt 0 , z gt 0 are respectively the 2^(nd) , 3^(rd) , 4^(th) terms of a G.P. and Delta=|{:(x^(k),x^(k+1),x^(k+2)),(y^(k),y^(k+1),y^(k+2)),(z^(k),z^(k+1),z^(k+2)):}|=(r-1)^(2)(1-(1)/(r^(2))) (where r is the common ratio), then

If |(x,y,z),(-x,y,z),(x,-y,z)|=k xyz , then k=