If `x,a,b` are real prove that `4(a-x)(x-a+sqrt(a^2+b^2))cancelgta^2+b^2`

If x is real, then the maximum value of y=2(a-x)(x+sqrt(x^2+b^2))

If the roots of the equation x^2+2c x+a b=0 are real unequal, prove that the equation x^2-2(a+b)x+a^2+b^2+2c^2=0 has no real roots.

If a+b+c=0(a,b, c are real), then prove that equation (b-x)^2-4(a-x)(c-x)=0 has real roots and the roots will not be equal unless a=b=c .

Assuming that x is a positive real number and a ,\ b ,\ c are rational numbers, show that: ((x^a)/(x^b))^(a^2+a b+b^2)((x^b)/(x^c))^(b^2+b c+c^2)((x^c)/(x^a))^(c^2+c a+a^2)=1

(x+iy)^(1/3) =(a+ib) then prove that (x/a+y/b)=4(a^2-b^2)

(x+iy)^(1/3) =(a+ib) then prove that (x/a+y/b)=4(a^2-b^2)

Prove that the product of the lengths of the perpendiculars drawn from the points (sqrt(a^2-b^2),0) and (-sqrt(a^2-b^2),0) to the line x/a costheta + y/b sintheta=1 is b^2 .

Prove that the product of the lengths of the perpendiculars drawn from the points (sqrt(a^2-b^2),""""0) and (-sqrt(a^2-b^2),""""0) to the line x/a costheta + y/b sintheta=1 is b^2 .

Prove that the product of the lengths of the perpendiculars drawn from the points (sqrt(a^2-b^2),0) and (-sqrt(a^2-b^2),0) to the line x/a costheta + y/b sintheta=1 is b^2 .

If x, a, b, c are real and (x-a+b)^(2)+(x-b+c)^(2)=0 , then a, b, c are in