Given x=cy+bz,y=az+cx and z=bx+ay, then prove `a^(2) +b^(2) +c^(2) `+2abc =1.

Let a, b,c be any real numbers, Supose that there are real numbers x,y,z not all zero such that x=cy+bz,y=az+cx and z =bx +ay . Then a^2+b^2+c^2+2abc is equal to :

If y = ax^(n+1) + bx^(-1) , then x^(2) (d^(2)y)/(dx^(2)) =

If a is the A.M. of b and c and the two geometric means are G_(1) and G_(2) , then prove that G_(1)^(3)+G_(2)^(3)=2abc

If zeros of a polynomial f(x) = ax^(3) + 3bx^(2) + 3cx + d are in AP prove that 2b^(3) - 3abc + a^(2) d = c .

Solve for x and y: ax+by=1" and "bx+ay=(2ab)/(a^(2)+b^(2)) .

If x-iy = sqrt((a+ib)/(c-id)) , prove that (x^(2) + y^(2))^(2) = (a^(2) + b^(2))/(c^(2) + d^(2))

If x + iy = sqrt((a + ib)/(c + id)) Prove that x^(2) + y^(2) = sqrt(( a^(2) + b^(2))/(c^(2) + d^(2)))

Prove that the circle x^(2) + y^(2) + 2ax + c^(2) = 0 and x^(2) + y^(2) + 2by + c^(2) = 0 touch each other if (1)/(a^(2)) + (1)/(b^(2)) = (1)/(c^(2)) .

If y=m x+c is a tangent to (x^(2))/(a^(2))+(y^(2))/(b^(2)) = 1 then b^(2) =

(i) If x-iy = sqrt((a-ib)/(c-id)) prove that (x^(2) + y^(2)) =(a^(2) + b^(2))/(c^(2) + d^(2))