If `|(a^(2),b^(2),c^(2)),((a+1)^(2),(b+1)^(2),(c+1)^(2)),((a-1)^(2),(b-1)^(2),(c-1)^(2))|=k(a-b)(b-c)(c-a)` then find the value of k.

Area of triangle whose vertices are (a,a^(2)), (b,b^(2)) and (c,c^(2)) is (1)/(2) . And area of another triangle whose vertices are (p,p^(2)), (q,q^(2)) and (r,r^(2)) is 4, then the value of |((1+ap)^(2),(1+bp)^(2),(1+cp)^(2)),((1+aq)^(2),(1+bq)^(2),(1+cq)^(2)),((1+ar)^(2),(1+br)^(2),(1+cr)^(2))| is a)2 b)4 c)8 d)16

If (x_(1)-x_(2))^(2) + (y_(1)-y_(2))^(2)=a^(2) , (x_(2)-x_(3))^(2) + (y_(2) - y_(3))^(2)=b^(2) , (x_(3)-x_(1))^(2) + (y_(3) - y_(1))^(2) = c^(2) and k[|(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|]^2=(a+b+c)(b+c-a)(c+a-b)(a+b-c) then the value of k a)1 b)2 c)4 d)none of these

Prove that |(1+a^(2)+a^(4),1+ab+a^(2)b^(2),1+ac+a^(2)c^(2)),(1+ab+a^(2)b^(2),1+b^(2)+b^(4),1+bc+b^(2)c^(2)),(1+ac+a^(2)c^(2),1+bc+b^(2)c^(2),1+c^(2)+c^(4))|=(a-b)^(2)(b-c)^(2)(c-a)^(2)

Prove that |(a,b+c,a^(2)),(b,c+a,b^(2)),(c,a+b,c^(2))|=-(a+b+c)xx(a-b)(b-c)(c-a).

The value of the determinant |(ka,k^(2)+a^(2),1),(kb,k^(2)+b^(2),1),(kc,k^(2)+c^(2),1)| is a)k(a+b)(b+c)(c+a) b)kabc (a^(2)+b^(2)+c^(2)) c)k(a-b) (b -c) (c - a) d)k(a + b - c) (b + c - a) (c + a - b)

Prove that Prove that Delta=|(1,1,1),(a,b,c),(bc+a^(2),ac+b^(2),ab+c^(2))|=2(a-b)(b-c)(c-a)

Show that abs[[1,a,a^2],[1,b,b^2],[1,c,c^2]]=(a-b)(b-c)(c-a)

If |(1,1,1),(a,b,c),(a^(3),b^(3),c^(3))| = (a - b) (b - c) (c - a) (a + b + c) , where a,b,c are all different, then the determinant |(1,1,1),((x-a)^(2),(x-b)^(2),(x-c)^(2)),((x-b)(x-c),(x-c)(x-a),(x-a)(x-b))| vanishes when a)a + b + c = 0 b) x = (1)/(3) (a + b + c) c) x = (1)/(2) (a + b + c) d) x = a + b + c